VOLUME I
GREEK, INDIAN AND
ARABIC LOGIC
Edited by
Dov M. Gabbay
John Woods
Handbook of the History of Logic
Volume 1: Greek, Indian and Arabic Logic
Handbook of the History of Logic
Volume 1: Greek, Indian and Arabic Logic
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Volume 1: Greek, Indian and Arabic Logic
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University of British Columbia
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King’s College London
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CONTENTS
Preface vii
Dov M. Gabbay and John Woods
List of Contributors ix
Logic before Aristotle: Development or Birth? 1
Julius Moravcsik
Aristotle’s Early Logic 27
John Woods and Andrew Irvine
Aristotle’s Underlying Logic 101
George Boger
Aristotle’s Modal Syllogisms 247
Fred Johnson
Indian Logic 309
Jonardon Ganeri
The Megarians and the Stoics 397
Robert R. O’Toole and Raymond E. Jennings
Arabic Logic 523
Tony Street
The Translation of Arabic Works on Logic into Latin 597
in the Middle Ages and Renaissance
Charles Burnett
Index
607
This Page Intentionally Left Blank
PREFACE
With the present volume, the Handbook of the History of Logic makes its first appear¬
ance. Members of the research communities in logic, history of logic and philosophy of
logic, as well as those in kindred areas such as computer science, artificial intelligence,
cognitive psychology, argumentation theory and history of ideas, have long felt the lack
of a large and comprehensive history of logic. They have been well-served since the early
sixties by William and Martha Kneale’s single volume The Development of Logic, pub¬
lished by Oxford University Press. But what such a work cannot hope to do, and does
not try to do, is provide the depth and detail, as well as the interpretive coverage, that
a multi-volume approach makes possible. This is the driving impetus of the Handbook,
currently projected to run to several large volumes, which the publisher will issue when
ready, rather than in strict chronological order. Already in production is the volume The
Rise of Modern Logic: From Leibniz to Frege. In process are volumes on Mediaeval and
Renaissance Logic, The Many-Valued Turn in Logic, and British Logic in the Nineteenth
Century. Others will be announced in due course.
As with the present volume, the Handbook’s, authors have been chosen for their capac¬
ity to write authoritative and very substantial chapters on their assigned topics; and they
have been given the freedom to develop their own interpretations of things. In a number
of cases, chapters are the equivalents of small monographs, and thus offer researchers and
other interested readers advantages that only a multi-volume treatment can sustain.
In offering these volumes to the scholarly public, the Editors do so with the conviction
that the dominant figures in the already long history of logic are the producers of the¬
ories and proponents of views that are possessed of more than antiquarian interest, and
are deserving of the philosophical and technical attention of the present-day theorist. The
Handbook is an earnest of a position developed by the Editors in their Editorial, “Co¬
operate with you logic ancestors”, Journal of Logic, Language and Information, 8:iii—v,
1999 .
The Handbook of the History of Logic aims at being a definitive research work for any
member of the relevant research communities. The Editors wish to extend their warmest
thanks to the Handbook’s authors. Thanks are also due and happily given to Jane Spurr
in London and Dawn Collins in Lethbridge for their indispensable production assistance,
and for invaluable follow-up in Amsterdam to our colleagues at Elsevier, Arjen Sevenster
and Andy Deelen. The Editors also acknowledge with gratitude the support of Professor
Bhagwan Dua and Professor Christopher Nicol, Deans of Arts and Science, University of
Lethbridge, and of Professor Mohan Matthan, Head of Philosophy and Professor Nancy
Gallini, Dean of Arts, University of British Columbia. Carol Woods gave the project
her able production support in Vancouver and is the further object of our gratitude. The
Engineering and Physical Sciences Research Council of the United Kingdom also sup¬
ported Woods as Visiting Fellow in 2000-2003, and for this the Editors express their
warm thanks.
Dov M. Gabbay
King’s College London
John Woods
University of British Columbia
and
King’s College London
CONTRIBUTORS
George Boger
Department of Philosophy, Canisius College, 2001 Main Street, Buffalo, NY 14208-1098,
USA
boger@canisius.edu
Charles Burnett
The Warburg Institute, Woburn Square, London WC1H 0AB, UK
charles.burnett@sas.ac.uk
Dov M. Gabbay
Department of Computer Science, King’s College London, Strand, London WC2R 2LS,
UK
dg@dcs.kcl.ac.uk
Jonardon Ganeri
Department of Philosophy, University of Liverpool, 7 Abercromby Square, Liverpool L69
7WY, UK
jonardon@liverpool.ac.uk
Andrew Irvine
Philosophy Department, University of British Columbia, Vancouver, BC Canada, V6T
1Z1
andrew.irvine@ubc.ca
Raymond E. Jennings
Department of Philosophy, Simon Fraser University, Burnaby, BC Canada, V5A 1S6
raymond.jennings@sfu.ca
Fred Johnson
Colorado State University, Fort Collins, CO 80523, USA
johnson@lamar.colostate.edu
Julius Moravcsik
Department of Philosophy, Stanford University Stanford, CA 94305-2155, USA
julius@csli.stanford.edu
Robert R. O'Toole
Department of Philosophy, Simon Fraser University, Burnaby, BC Canada, V5 A 1S6
Tony Street
Faculty of Divinity, University of Cambridge, West Road, Cambridge, UK
ads46@cam.ac.uk
John Woods
Philosophy Department, University of British Columbia, Vancouver, BC Canada, V6T
1Z1
jh woods ©interchange.ubc.ca
This Page Intentionally Left Blank
LOGIC BEFORE ARISTOTLE:
DEVELOPMENT OR BIRTH?
Julius Moravcsik
INTRODUCTION
“What is the origin of logic as a distinct discipline?” is a complex and partly con¬
fusing question. It is confusing because some might misinterpret it as asking for
a date at which people discovered the difference between sound and unsound rea¬
soning. But presumably people have been thinking logically, at least in relatively
simple contexts, since the origin of humanity. Material elements have behaved
“physically” much before the rise of physics as a discipline, and people, at times,
have argued logically much before the first system of logic was presented. There is
a difference between the two cases. Physics did not start with everything “physi¬
cal” beginning to think about what normal physical functioning is. Humans, with
a certain sense of detachment, started raising that question. Reason had to be
applied to natural processes in space and time. In the case of logic, however,
reason had to be applied to reason. This application required that people reflect
on their own thought processes and that of others. This reflection then had to be
coupled with separating the art of logical reasoning from other subjects. However,
this separation was not like the separating of two natural sciences, e.g., chemistry
and biology. “Separation” in our case has two aspects. 1 We need to separate logic
from other disciplines dealing with argumentation and communication, such as
rhetoric, advertisement generation, and others. But there is also another sense.
For in the case of logic we need to bring our reflection on language to a new,
higher level of abstraction. We need to consider language, like mathematics, as
an abstract system, and then isolate a higher level abstract quality, namely valid
and invalid inference discriminability. Language in practice is a series of sounds.
We next abstract from that the phonological, syntactic, and semantic elements.
We then consider the grammar and semantics, and attempt to impose on some
of this logical structure. So we delineate the valid inference patterns. To justify
this effort we need to bring in rules of valid inference. But these do not consist
of independent elements. Logic emerges when we can relate our rules of inference
and present them as a coherent system. Finally, we reflect on the abstract features
of such systems in order to understand what logic itself is. Nobody has come up
'.Metz, R., 1999, The Shapiiig of Deduction in Greek Mathematics, (Cambridge UP), chapter 1.
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
(c) 2004 Elsevier BY. All rights reserved.
2
Julius Moravcsik
with an informative and explanatory definition of what logic is. The same holds
for mathematics. Logic and mathematics evaluate sequences of elements from a
completely detached view. By this we mean that the point of view is not influ¬
enced by considerations of utility, pleasure, and other such human interest-relative
factors.
It would be misleading to characterize the emergence of the first system of
logic as either an invention or a discovery. It is not an invention like artifacts,
such as wheel, car, or less concrete, such as an alphabet. On the other hand,
it is not like the discovery of a mountain nobody knew about, or a chemical
element. There are elements of both invention and discovery in the formulation
of a logic. We discover necessary relationships between abstract elements that
can be characterized linguistically or conceptually. We have, however, options as
to how to characterize these relationships. At that point, inventiveness enters the
scene. We cannot credit the discoverer of a river we did not know, with originality.
Nor do we credit a logician with originality when he first presents the law of non¬
contradiction. But we do credit him with originality in view of the particular ways
in which he shapes rules for deduction and proofs.
If we think — as we should — of logic as a system of justifiable rules of inference
that demarcates the valid from the invalid ones, then we should say that Aristotle
was the first creator of logic in Western culture, and that this achievement came
spontaneously. As Carl Hempel used to say, it was “a free creation of the human
mind”. It took a special leap in cognition to arrive at this high level of abstractness,
and to formulate the laws of syllogisms the way Aristotle did. On the basis of these
considerations one can say that there was no logic in the West before Aristotle,
and that emergence was spontaneous, not a matter of gradual development.
Nevertheless, we should not think of the rise of logic as taking place in an
intellectual vacuum. Logic presupposes a set of concepts that provide its back
ground and elements. These concepts are interrelated. 2 They do not surface
isolated from each other. Their respective developments need not be interrelated,
but the final products must be linked that way.
In the following we shall trace the developments of some of these concepts. We
must not think that once these concepts are parts of a culture, that logic must
emerge. The conceptual background we will trace constitutes a necessary but not
also sufficient background for the kind of ingenious work Aristotle did.
Thus we divide the question: “was there logic before Aristotle?” into two sub¬
questions. First: “did anyone produce a system of logic comparable to that of
Aristotle prior to the work of the Stagirite?” and “what, in any, jumps in level of
abstraction, and development of new concepts were required for the rise of logic?”
The answer to the first question is negative. To the second question we respond
by showing what levels of abstraction were needed, and how concepts developed
that formed the needed background for logic. In the next section we turn to a
brief sketch of this conceptual background.
2 Ibid, chapter 3.
Logic Before Aristotle: Development or Birth?
3
1 DEVELOPING THE CONCEPTUAL FOUNDATIONS
The conceptual foundation has two aspects. One of these is the development of the
needed detachment and objectivity. One can assess arguments in a number of ways.
For example, how persuasive these are, how colorfully they are presented, how easy
it is to understand them and so on. We need to eliminate such considerations when
we assess something as valid or not. Thus we should trace the development of the
notions of generality and objectivity. Detailed work in this area is yet to be done.
We need to trace the move from “this must be right because the gods say so” to
“this must be a sound deduction because each step can be justified by the rules of
inference.” In general terms, the tracing of this development was started by Bruno
Snell, with the felicitous title to one of his chapters, “from myth to logic” . 3
Before we start on the details, it is worth clarifying where we should look for
relevant evidence. For example, the Kneales suggests that we should look primarily
at texts of geometry and the natural sciences that started around that time. 4 They
suggest that literary sources are less likely to contain logically sound texts. But it
seems that if we regard the use of logic as a universal human practice that is to be
codified, then we should look at literary texts as well There are many arguments
in Homer’s Iliad. The problem lies not so much with the deductive links but with
the premisses; clearly a matter outside the province of logic. Many would not
accept the premiss that everything that is loved by the gods is necessarily good,
but whether the other beliefs that surround this one are consistent with it or not is
a matter of logic. Some discussions involve sketching alternatives among which the
characters must choose. For example, who should have more prestige and power
the tribe, the one of royal blood, or the most successful warrior in the tribe? 5
Some notion of what argument supports what — apart from what one would like
to be the case — seems implicit in the exchanges. Such material can be seen as
leading to the development of the notion of consistency. Some might be tempted to
compare agreements, and see something more general and abstract that these have
in common, even if this cannot be yet articulated as logical consistency. For the
latter concept to be used, one needs the notion of logical form, more abstract than
anything we need in other disciplines except mathematics. We have no idea what
enables humans to discern the same “form” in so many arguments with different
vocabularies and dealing with different topics, different domains. Some suddenly
found the similarities, on this abstract level, puzzling. This puzzlement need
be resolved; hence the notion of logical form. To take these conceptual steps may
seem to many today easy. This would blind us to the difficulty with which different
cultures sooner or later managed to form the right notion. A good comparison
would be that with grammar. Logic needs some sort of a grammar for the internal
analysis of sentences. Yet the first formal complete logic was formulated only in
3 Snell, B., 1953 The Discovery of the Mind, tr. T. Roseuirneyer, (Harvard UP), p. 9.
‘'Kneale, W. and M., 1962 The Development of Logic, (Oxford UP), p. 1.
5 Homer, Iliad, Bk.I., lines 180- 280.
4
Julius Moravcsik
120 B.C., quite a bit later than Aristotle’s activity. 6 Yet Aristotle does rely on
grammatical distinctions and composition rules.
The logical form of a sentence explains some of the semantic features of that
unit. And yet the relation between logical deduction and explanatory power is
complex. Part of the problem is that there is no complete agreement on what
explanatory power in general is. Some think that this can be explained within the
formalisms of today’s logic, 7 while others deny that. 8 So one could point to the
understanding of a mathematical proof as a holistic cognitive phenomenon, going
beyond mere understanding of logical relations.
It is safe to say that not all deductive patterns have explanatory power. Our
drive towards logic is not based solely on finding sound patterns for explanation.
Logic is also involved in human efforts to prove something, to sustain an argument
in debate, and to find an unassailable stand which nobody can legitimately attack.
There is no one drive for logic. The motivation is pluralistic.
In Aristotle’s way of thinking there is clearly a strong link between explanation
and demonstration and hence also deduction. All men are mortal, all Greeks are
men, therefore all Greeks are mortal. The term “men” carries the key explanatory
power in this, configuration. It is through being human, of having human nature?
that the Greeks have their mortality.
Yet Aristotle himself points out that the explanatory power does not come from
the form alone. As he says, one must select the “middle term” — “human” in our
example, very carefully. 9 There will be a number of co-extensive middle term
candidates; so how do we choose the “right one”? Aristotle thinks that this is
a matter of insight; we need intuitive understanding of what is puzzling us in a
context to see what “really” explains.
So explanation and deduction overlap. Some explanations are not deductions,
and some deductions are not explanations. But the overlap is important, or at
least it seemed to Aristotle the most important part of our rational activities. His
predecessors did not link the two, but Aristotle saw a way in which — he hoped
— one could.
One of the expressions the development of which led to the notions of evidence
and thus also to premiss, is “signs”. Signs show in Homer what the gods want,
signs suggest happiness or tragedy, signs signal to Priam that he can trust, at the
end, Achilles.
Once we trust regularities in nature, signs need not be arbitrary. We use these
in order to predict storms and other weather conditions. Interestingly, both of
these uses indicate something that will give us — or so they thought — certainty,
but for quite different reasons. In one case the certainty is derived from religious
thinking. The source of the certain, the assurance, is beyond typical rational
6 Dyonisius Thrax
7 Salmon, W., 1989, “Four Decades of Scientific Explanation” in Minnesota Studies in the
Philosophy of Science , (Minneapolis University of Minnisota Press) pp. 3-196.
8 Manders, K., “Diagram Contents and Representational Granularity”, mimeographed paper,
University of Pittsburgh Philosophy Department.
9 Aristotle, Post. An. Bk II.
Logic Before Aristotle: Development or Birth?
5
understanding. In the other case the source is the regular observation of general
truths in nature. In modern times we link the predictability of rain, storm and
other such phenomena to probabilistic reasoning, but it was not construed that
way in ancient times. This turned out well in a way, for in this development kinds
of certainty paved the path for forging the necessity — and correlated One might
think that the sciences that promise only probability developed first, and the ones
bringing certainty only later. But exactly the opposite is the case. Mathematics
and geometry had their early flowering and the “natural’ sciences only later.
One might think that the development was “from probability to certainty”, but
in fact the reverse took place. Humans reach for what promises certainty and settle
for the probable only when the methodology and selection of the right domain of
objects is not at hand.
We shall now look into the developments, using Bruno Snell’s work, the title of
which we cited already. In the light of the last paragraph we can see now myth
and logic not only as terminus ab quo and terminus ad quern, but also as sharing
an important characteristic, namely the promise of certainty.
There is not enough evidence to trace out the separation between the two sources
of certainty. In many countries the two live side by side, and in Plato’s Timaeus
the rough equivalent to some of what are today natural sciences is introduced
within a religious framework. One can speculate about how questioning of a well
known sort, namely, going from the particular to the general helped to sort out
different kinds of certainty. In the one case, people presumably started to ask
questions about the reliability of specific divine commands and alleged forecasts.
But after that they could ask questions about the nature of these pronouncements
in general. For example, an important critic, Plato, treats them as a group, but
here generality is not available. Divine orders, as also divinities, remain particular.
On the other hand, after we agree on specific rules of deductive inference for some
patterns, we can ask for more general justifications, and a system of logic will
provide this for us.
In our tracing the development of what was necessary background for logic we
will side-step the question of whether at earlier stages logic was seen as necessary
rules of thought, or necessary rules about how elements of reality function, or an
autonomous discipline dealing with its own unique domain. For these are meta¬
physical questions, and do not touch on the nature of early systems of logic as
general theories about validity. Aristotle’s work suggests that he saw logic as hav¬
ing both metaphysical grounding and reflecting necessary features of thought. 10 It
is reasonable to suppose that the first clear examples of demonstration and proof
in the Greek world came from geometry, or rather what was then the combined
subject of mathematics and geometry. Here abstraction and rigor combine and
reach the same high level that logic does. 11 Nevertheless, we must not make the
mistake of thinking that logic grew out of the practice of geometry and mathe¬
matics. For one can conceptualize geometrical demonstrations as limited to the
10 Aristotle Metaphysics , Bk. Gamma.
n Netz, R. Op.cit ., in general.
6
Julius Moravcsik
particular domain of mathematical and geometrical entities. “He need not look
at reason having a universal domain in order to do rigorous geometry. This does
not mean that geometry might not have influenced Aristotle in his construction of
logic. In fact, there are signs suggesting that Aristotle had an independent con¬
ception of logic, but wanted logical demonstrations to be in some ways analogous
to geometrical demonstrations. 12
Thus we can see that neither grammar nor geometry should be interpreted as
the forerunners of logic. As we say, the first grammar was constructed by Dy-
onisius Thrax around 120 B.C. — quite a bit after Aristotle’s logic. But apart
from the temporal issues, we can see why neither of these subjects provide all that
logic presupposes. Geometry does not because its methodology is not sufficiently
general, and grammar not, because though it moves on the required level of ab¬
straction, and is sufficiently general, it lacks the rigor at least in earlier times that
logic requires.
In the case of geometry the interest in explanatory power and in demonstration
comes happily together. Furthermore, geometry can be seen also as a paradigm
for at least some types of deductive reasoning.
In summary, then, we can interpret the three interests, in explanatory power, in
proof and deductive reasoning and argument assessment, as stimuli for the devel¬
opment of that set of concepts. Thus the vocabulary that is necessary for a system
of logic came into being. Analagously we can speculate that the combination of
these interests would motivate people to work toward the development of the log¬
ical concepts and vocabulary. Thus the three interests can be seen as underlying
logic, and psychologically, as underlying human efforts, conscious or otherwise for
formulating logic.
2 CONCEPTS AND VOCABULARY PRESUPPOSED BY LOGIC
We have sketched the salient interests that would lead people towards constructing
a system of rules generating and assessing logical validity. We now turn to the
vocabulary that logic requires.
First we examine the notion of truth. Truth is clearly needed, for without it we
could not articulate notions like premiss, conclusion, and consequence. We must
assume that grammar already provided the notion of a sentence, and thus we can
understand the way in which truth is attached primarily to sentences. Truth is
also what we need as a contrast to falsehood, a notion to be discussed later.
Presumably some notion of truth existed since the dawn of human history. It
may not have been separated from some general notion of what it is to describe
something correctly. Furthermore, in its early forms truth was not separated from
what is true in evaluative ways. Something can be a true or genuine diamond,
friend, a true alumnus, a genuine Egyptian artifact, and so on. The evaluative
aspect emerges in contexts in which we wonder whether to apply this term to a
12 Aristotle, Prior Analytics.
Logic Before Aristotle: Development or Birth?
7
friend or mere well-wisher. One can only speculate on whether there was a notion
of a true sentence as a truly genuine real sentence, i.e. one that did its job and
gave a good representation of a part of reality in which we are interested.
In any case, we need to abstract various aspect of this notion of true F, or real
F, in order to work towards forging the truth that logic wants. First, we need to
take away the positive evaluative aspect. A true proposition or sentence may be
bad news, or describing evil doings. Secondly, we need to change the gradational
aspect of truth into a non-gradational one. What is genuine can be a matter
of degrees, and the same holds for a true friend or true spring weather. But a
sentence cannot be really true or not really true, or just half-true. If it were of
that sort, it could not do the job that logic demands of it.
We need now a further level of abstraction. We use descriptions as good for
a certain community. Descriptions function in contexts and with qualifications.
Some of these are relational, others introduce pragmatic contexts of description.
All of this applies especially to nouns designating artifacts like ‘table’. How much
damage can an object endure and still qualify to be a table? What is a table for
a certain community need not serve as such for others. But when the word ‘table’
occurs in a logical construction we abstract from all of this. Either there is a
table or not, and either it functions appropriately in an intended premiss like “all
tables are...” or should be replaced with another equally context and gradation
independent term (noun, verb, adjective).
We are still not quite finished with our account of “assent to truth as used in
logic”. Plato has a characterization of truth in the Sophist, 13 and there are indi¬
cations that the formula comes from earlier times, “the true sentence expresses
things that are, as they are”. The emphasis of the sentence being about and de¬
scribing real things need be taken away if we are to see “true” as a purely logical
notion. But apart from that, there is the promising but troublesome expression”
as these are”. We need to give this an interpretation that transcends the differ¬
ences between philosophical theories of truth, such as correspondence, coherence,
pragmatic, redundancy etc, “theories”. As Tarski noted the logician’s notion of
truth is independent of all of this. Furthermore, it is a notion that can charac¬
terize sentences in systems the domain of which may turn out later to be seen as
illusory. 14
Plato knows that he is not offering a reductionist definition of truth. Nor did
Aristotle attempt such an account. Once we reached beyond all of the abstractions
listed above, we van only say: “what is left” is the truth required for logical
constructions and inferences.
It is interesting to ponder the two very different views that emerged concerning
this “ascent”. According to one view, Platonic in origin, we “purify” language
and our concept of truth as we reach the level of abstraction needed for logic.
Purification is no longer a much used concept, but the modern term idealization
I3 Plato, Sophist, 263b-c 1936.
14 Tarski, A. “The Concept of Truth in Formalized Languages”, in Logic, Semantics and Meta-
mathematics , 1956. pp. 152-278 (Oxford: Clarendon Press).
8
Julius Moravcsik
will do just as well. According to the alternative view, we oversimplify meanings
and deprive the term of all of its richness when we restrict it to the use needed
for logic. On the one hand, one can argue that without the restriction no logic,
no great expressive power. On the other hand, one can argue also that with the
abstractions we lose a lot of the flexibility, metaphoric power, simple, and other
such literary devices that enrich languages so much. It is quite wrong to think of
these devices as just decorative elements. They carry meaning, help to think about
the more indeterminate aspects of what we talk about, and are very important as
vehicles for gradual changes of meaning for words either in scientific or everyday
or literary contexts. It is an interesting peculiarity of natural languages that we
cannot have it “ both ways”. Thus in our uses of language we choose to stress in
some contexts this and in others that aspect of meaning.
It is natural for us to turn now to another important concept of language with¬
out which logic cannot be conceived, namely that of negation and falsehood. These
are distinct concepts, but at times their extensions overlap. The notion of truth
makes no sense without a notion of falsehood. Falsehood could have well origi¬
nated in connection with normative notions like honesty. In Sophocles’ Philoctetes,
Odysseus is trying to persuade the young Neoptolemus to lie to Philoctetes. Lying
must carry, at least implicitly, the notion of falsehood, for presumably to lie is not
to tell the truth; “the way real things actually”. Successful prediction is also an
ancient notion, whether in connection with the diumation of priests or weather
forecasts (these two might overlap). So we can look at various practices such as
being honest, being good at forecasting weather, or not, and from such notions
abstract the notion of falsehood. The failures of practices like the ones mentioned
would — on detached analysis — yield the notion of falsehood. It is impossible
for us to construct what would be by even lax standards reasonable hypotheses as
to when these abstractions became explicitly the objects of cogitation. Falsehood
must have been an essential ingredient in the conceptual framework within which
mathematics and geometry were practiced as sciences and so conceived consciously
by the practitioner. But even so, it takes an additional step of abstraction and
generalization to extend the notion of falsehood to assessment of descriptive speech
in general.
As we turn to negation, we must draw an important line between that notion
and some others following in our discussion, and truth and falsehood. For truth
and falsehood are not conceived at any stage in history as forces of nature. These
are not metaphysical concepts. But negativity and one thing following another
are. (If so the one points out that much later for Frege the True and the False are
objects, one should separate this purely abstract notion of objectification, installed
as a part of a highly abstract system of semantics from the kind of metaphysical
or natural posits about which we talk here.)
In tracing the notion of negation, we should start with the notion of opposites,
in particular opposing natural forces like fire and water (Heraclitus), or moistness
and dryness. Was their incompatibility construed as necessary and a, priori, or as
just an extreme case of clashing natural forces, is an unanswerable question.
Logic Before Aristotle: Development or Birth?
9
From the notion of opposing natural forces to the logical notion of negation,
one has to climb a long and steep road. First, opposites need not exclude each
other completely. The weather maybe “stormy” with what the British call so
characteristically — “sunny intervals”. Or it could be between stormy and calm.
With hot and cold, a thing can be hot in some respect, and not hot in others.
By the time we see opposition illustrated in Plato, we come to examples like tall
and short. These are what are called later qualities. Furthermore, semantically
they behave like adverbials. A tall monkey may be a short jungle-living animal.
The distinction between natural forces and the more quality-oriented classification
is not an all-or-nothing affair. What about light and darkness? Whether one
regards these as forces or not, depends on one’s physics, rudimentary as this may
be.
Plato wrestles a lot with the “negative”. His thinking about negation is strongly
influenced by the Parmenidean attack of this as not-being. This explains also
why his first attempt of characterization not in terms of sentence-negation, but
predicate negation. He is anxious to point out. 15 that the not-fine is as much a
part of reality as the fine. Fine things and not-fine things are different, but both
existing. Furthermore, the difference is a special “contrast” that Plato does not
define any further. Nor has anyone since. Hence the problem is swept under the
rug. In what consists the negativity of the not-fine? Otherwise? This is ???.
One cannot insist that we arrive at a completely adequate account of negation
and falsity only when e.g., negation is completely divorced from metaphysics. For
example, a constructive step forward is the realization that the completely nega¬
tive (predicationally) entity is conceptually impossible. So the negative concerns
always some aspect of what we talk about. With negation we indicate in a unique
way that what we talk about is different from that which the corresponding posi¬
tive description would represent. The early treatments of negation in metaphysical
and ontological terms is responsible for the early concentration on predicate nega¬
tion, before subsequent consideration of what is logically prior, namely sentence
negation.
Some might regard it as a step forward when negation is considered a purely
logical operation of contrasting what is taken in a language or system as positive
with the negative. But Aristotle himself can be certainly described of having a
system of logic. Yet he could assign in the case of predicate negation an important
difference between the positive and the negative. According to his view a negative
predicate, e.g., not-human, was indefinite. 16 Given what we know about Aristotle’s
views about unities of predicates and their significance, we can represent this
Aristotelian view in the following terms. A positive predicate like “is a human”
(or in term logic “human”) has a unity that is seen by considering the conceptually
related principle of individuation; if we understand what ‘human’ is then w also
understand what it is to be 1, 2, 3, etc. human(s), even if in some concrete
15 Plato, Sophist, 258-259.
16 For a discussion of this notion see Thompson, M. “On Aristotle’s Square of Opposition”,
Philosophical Review, 1953 pp. 251-265.
10
Julius Moravcsik
context the counting is difficult to carry out; e.g. battle fields. But the predicate
“not-human” has no individuation principle attached to it. In principle we should
be able to answer the question: “how many humans in this room?” but there is
no correct answer in principle to the question: “how many not-humans in this
room?” We can count the not-human in an infinite number of ways, each equally
good or bad. In this respect positive and negative predicates differ. (This does
not interfere with the logical operation of double negation yielding a positive.)
We shall now turn to the last of the pillars that is needed in order to have a
conceptual framework within which logic can be conceived. This is the notion of
“p following from q ” where ‘p’ and ‘ q ’ represent descriptive sentences. In short, we
need the notion of logical consequence. How does language build up this notion?
We shall show in terms of the key words used gradual emergence in ancient Greek
of the needed notion.
First we shall consider the word “akolouthein”.
In its earlier less abstract uses it means following someone in a general physical
sense; for example, soldiers following others in rows. The stress from our point
of view is not merely the concrete domain of application, but also the element of
order associated with the term. When armies are set up, soldiers and their rows
are occupying designated places, and traverse designated routes. Thus it is not
surprising that we find also usages in which the word denotes natural phenomena
following each other such as cloudy sky followed by rain. 17 Here the notion of order
has more force. The “following” is a matter of the laws/regularities of nature.
We find also uses in which the word stands for guidance and obedience. In
each of these cases, the key force is not just sequencing, but things following each
other because of natural order or rational human order (the wiser, or in position,
demanding obedience from the lesser.)
Finally meaning is raised to an abstract level, and our word designates sequence
in argument. This is abstract but too wide and not sufficiently structured. Once
the final stone in the diadem: x following logically from y. In other words, the
notion of logical consequence. Hence a key cog in justifying inferences.
The other expression Aristotle uses in this connection is “sumbainein”. This
means originally “standing with feet together”. But other senses emerge, such as
joining something, and come to agreement. Thus in this case too we see both
movement towards abstract levels and differentiation of ingredients. Things are
joined according to a certain order, and their “agreement” signifies harmony of
elements. We see also the use of this word for consequence, and necessarily joining
things. Eventually the word denotes inevitable sequences, and thus becomes a fine
vehicle for Aristotle to designate logical consequence.
We have, then, key ingredients in the conceptual framework within which Aris¬
totle’s syllogistic logic was formulated. We turn now to a basic notion absolutely
necessary for explicating logical relations, namely predicates or terms (for our
purposes we will not need to make here fine distinctions.) The basic structure of
logical formulae in modern symbolic logic is the same as in the logic of Aristotle.
17 Snell, Op.cit ., p.212.
Logic Before Aristotle: Development or Birth?
11
In the premisses and conclusions relations between predicates or terms are rep¬
resented. First let us see on what level of abstraction we need to construe these
terms. In a sentence like “All A’s are B' s” the A and the B need be taken as
having potentially: Universal application; must be independent of subject matter;
and should be precisely delineated, without polysemy or ambiguity.
We need to deal with two further factors, if only to lay these aside: First, it is not
relevant to their employment in schemes of the sort just indicated that our speci¬
fication or of the content of the terms is in most cases dictated by human interest,
bias of our perceptual system, etc. One can reason logically with pure concepts,
detached from human interest and with ones reflecting bias. This will not affect
what is called logical form. Secondly, this characterization of terms/predicates is
neutral with regard to ontology. The Platonist and the nominalist will have to
present both an interpretation of “A” and “B” that allows these expressions to
figure in the purely logical characterizations of “some A” or “no B".
All of this may sound trivial to a philosophic or mathematical audience, but we
must make a real effort to try to imagine a world of ideas in which these levels of
abstractions are not yet present and are not picked up by adequate vocabulary.
And yet, it is important to stress again that we are not accusing the first humans
to have had a materialist bias. Rather, they used languages in which many distinc¬
tions fundamental to the delineation of logic have not yet been made. The fact
that the abstract has not been separated from the non-abstract does not mean
that each relevant word had only concrete entities in its domain of designation.
Undoubtedly, there are many ways of sketching speculatively the development
of the terms of logic. In the following, we rely heavily on the scholarly work of
Bruno Snell.
Any natural language with sufficient communicative power to serve as describing
reasonably vast areas of reality must contains words of general power, and hence
words describing things with oversimplifications. Thus the meanings of ‘lion’ or
‘horse’ ignore all of the specific differences between specimens within the respective
species. This is governed by two conditions. First, separate words for every
qualitative difference between specimens would create languages with absurdly
large vocabularies. Secondly, the ignoring of specific differences is dictated by the
needs of the projected linguistic community. As is well known, some of the tribes
in northern regions have many words for different kinds of snow. This is because
these differences play roles in the securing of practical necessities in their daily life.
Thus modes of life-style at times push towards oversimiplifications and at other
times towards generating many senses for the same word.
Snell discusses in the early parts of his “from myth to logic” natural kind terms
like ‘lion’ and ‘horse’. 18 That does not mean that he thinks of these as the earliest
words, but that these are the kind of word (noun) that plays crucial roles in the
development of the notion of predicate. Brief reflection should show us why this
is the case. With kind-nouns like “lion” and “horse”, application is an either-or
proposition. We will not say things like “akind of lionish thing”, or “more or less of
l8 Ibid. pp. 201-207.
12
Julius Moravcsik
a horse” (mythical entities excluded). But when we turn to verbs we find a different
situation. Is someone walking? There are clear cases showing the affirmative, and
on the other hand, clear cases of being in a stationary position thus deserving
the negative. But in between there are many cases that can be interpreted either
way. Furthermore, there are different criteria for walking depending whether it is
supposed to be an event of walking for a baby, a normal healthy adult, a recovering
patient, or a moon-walker. Similar cases can be shown for most other verbs raining,
singing, dividing, melting, boiling, etc.
In order to use verbs as predicates or subjects in syllogistic logic we must ide¬
alize, and abstract away from in-between cases and relativity to subject. At the
same time, we must admit that verbs with their cases and argument places provide
most of the structure of the sentence. Thus what is fundamental from the point
of view of constructing a logic is not fundamental from the point of view of expli¬
cating what gives the sentence its basic syntactic and thus also partial semantic
structure. These reflections should not lead us to the conclusion that verbs are
intrinsically vague and plagued by polysemy, and cannot be sharpened in their
meanings to serve as terms in syllogistic reasoning. The point is, however, that
we must do a lot of abstracting before they can serve this way. In the case of the
types of nouns we considered we need to abstract away from differences between
individual specimens of a species, but these are all on the most elementary level
of abstraction, and easy to ignore.
We do find verbal meanings sharpened in certain contexts in which the verb
with many others is used within a certain practice, or applied jointly with others
to illuminate a certain domain of entities. For example, in mathematics, there
is no room for considering many senses of ‘divide’. The relevant meaning is that
which we use in connection with the mathematical notion of ’division. (The same
holds for ‘addition’, subtraction’, etc.) Thus we can say that one should look
for certain disciplines or domains with respect to which certain words, or even
word families, need to have sharpened meanings. Such might be: mathematics,
geometry, chemistry, and other sciences, or applied fields of the sort we today
call engineering. But we should not think of precision limited to the sciences.
Certain financial transactions and other economic exchanges, once precise units
of what serves in the exchange are determined, require also precise meanings (i.e.
sharply delineated (ones). So one might conjecture that we build logic once we
have become accustomed to think within certain fields very precisely. But this
might not be right. For what logic requires is precisely to think universally, to
understand logical form regardless of subject matter. This contrasts with the
restricted rigorous thinking of geometry or certain parts of economics. Logic is a
set of rules of inference which we judge to be within valid patterns, regardless of
what the ”A’s” and “B’s” stand for.
Let us, then return to consider the development of other parts of the vocabulary,
and the devices with which we can raise the semantic content, and thus the thinking
with these concepts, to higher levels of abstraction.
It might be thought that the kind concepts we form are based on resemblences
Logic Before Aristotle: Development or Birth?
13
we notice. Since there is an infinite number of these in any context, we should add
“salient” to this proposal. But even with the emmandation it seems weak. For
what unites our concepts of both human creations and many natural kinds that are
in some way related to human interest is their functional aspects. We call many
things that are quite different in shape, material constituents, and aesthetic looks
houses, because these buildings offer shelter and a place where what in a specific
context can be regarded as suitable for carrying out usual human functions can
indeed be carried out.
The problem is that the more functional a concept is, the more likely it is to
be subject to vagueness and polysemy. It is no surprise that Aristotle thought
geometry to be providing good examples of deduction. Abstract generality, no
vagueness, are inherent in geometrical notions. The problem we are discussing is
how to carry out high level abstractions in order to have all or most parts of a
natural language be capable of producing elements that can functions as terms in
syllogisms.
There are some kinds of cases where Snell is right and resemblance rules, but
that is because we supplement similarity with a “built-in similarity space”, as
Quine pointed out. 19
In the case of terms like ‘horse’ and ‘lion’, we are in a fortunate semantic sit¬
uation. On the one hand, the terms by themselves can function well in logical
inferences, once we abstract the layers of meaning relating to human interest and
differences between specimens. On the other hand, from an early stage of develop¬
ment on, we used these nouns with the equivalents of ‘...like’ (as in ‘lion-like’) to
set up what seemed to the linguistic community salient similarities, with the exact
nature of the similarity left open. In this way on could use solid “noun-blocks”
and with the additions give flexibility, room for further development, and beauty
to language. Consider “white as snow”, or “sweet as honey”, used in Homeric lit¬
erature. Maybe if they had discovered something sweeter than honey, they would
have used another comparison to describe the objects under consideration. So the
development of the descriptive vocabulary moves on from comparisons to simile
to metaphor, and that, in turn can become “calcified” and turn into a noun or
adjective with precise meaning. Much technical vocabulary develops this way,
also in philosophy. E.g., Plato uses the Greek equivalent to “partaking” to mark
the unique indefinable relation between Forms and what modern philosophy calls
instances. What starts as a metaphor becomes a part of Plato’s technical vocab¬
ulary.
Another important way in which abstract thinking develops even though we
do not tend to think of it this way is by analogies which, when sharpened, can
be restated in terms of proportions. We live in cultures in which basic units of
measurement for length, area, time, and three-dimensional content are taken for
granted. This separates us sharply from the Greek culture in which such exact
units were not available for a long time. Thus the basic notion for measurement
was proportion. For example, — though it is hard for us to fathom this — the
I9 Quine W.V., Word and Object, (John Wiley, New York) 1960. p. 83.
14
Julius Moravcsik
Parthenon was built using constantly proportional instructions for building.
In order to raise this mode of thinking to how we would understand mathematics
proper, and with this the production of the mathematics of exact measurement,
we need to focus on how proportions can be raised from the purely empirical to
the mathematized.
Not all proportional thinking can be handled in this way. For example, as
Snell notes, in the Gorgias Plato set up the following proportion. 20 Rhetoric is to
philosophy as cooking is to medicine. The proportional statement is articulated
in order to shed light on philosophy. Of course, in this form whatever conception
of philosophy emerges, it cannot be used in syllogistic thinking. But the problem.
of making the characterization formulated by the proportional statement explicit
and sharp, fuels the mind sooner or later to try to characterize all of the terms
in this statement in explicit and unambiguous ways, so that Plato’s conception of
philosophy could play roles in further deductive arguments. One might think that
in order to shape an acceptable vocabulary for logic, one must take as fundamental
those attributes that are quantitative in character. But this does not follow. The
concepts must have principles of individuation. We understand what ‘horse’ means
when we know how to count horses. But that does not make the meaning of ‘horse’
quantitative in the sense in which what linguists call “mass terms”, such as terms
for colour, smell, weight, etc., quantitative.
These reflections show how much work must go into reaching the level of ab¬
straction at which the relevant concepts and terms needed as the background for
logic, can emerge. As said before, merely having the vocabulary for logic does not
necessitate the emergence of a system of logic. But in this case a drive for more
generality in explanations, uniting criteria for good proofs, arguments, and some
kinds of explanations, forces the mind also towards higher levels of abstractness.
Hence the idea of a system of logic in which the rules form nets of justification,
and subsequently, the reflections of what different systems of logic might have in
common. In this way we arrive at the notion of theory of logic. I do not think that
Aristotle reached this level, but dealing with that thorny issue is someone else’s
job.
Before we leave this developmental sketch we need to say something about quan¬
tifiers. These are essential parts of the logical vocabulary. On the other hand, the
needed words do come from ordinary language, and the changes needed for these
to turn into technical expression are not formidable. Let us take the existential
quantifier “some”. Most of our everyday uses of this term are contextual; some
students, some luck, some countries, etc. But even common sense flirts with the
use of ‘some’ that is designed to cover all of reality, such as the claim: “there are
some things money cannot buy.” But note that this use requires generality only.
It does not require a leap to a higher level of abstractness. One might insist that
the logical use of ‘some’ entails thinking of the range of application all possible
entities, abstract or concrete. But such a range is neither officially acknowledged
nor excluded from the everyday use.
20
Snell, Op.cit., p.221.
Logic Before Aristotle: Development or Birth?
15
We find the same situation with regard to the universal quantifier. Zeus is
described in one of the Greek dramas as “the all-conquering” one. This hardly
specifies in a sharp way what ‘all’ covers, nor does it rule out anything. To obtain
the logical sense all we need to do is consider the religious sense of ‘all’ as in
mythology, and sharpen the delineation of the domain. In this interesting way,
one can think of aspects of religion as preparing the way to logical vocabulary.
After this positive sketch of how the background of logic requires development,
even if the first system of logic does not, we will turn to devices that some historians
have wrongly taken to be forerunners of logic.
3 LOGIC AND DEFINITIONS
In their influential book on the history of logic the Kneales say about Plato that
“he is undoubtedly the first great thinker in the field of the philosophy of logic. He
treats... three important questions that arise as soon as we begin to reflect on the
nature of logic...”, and he thinks one of these is “what is the nature of definition,
and what is it that we define?” 21 In this brief section I would like to show that
this view is mistaken, and that in fact, whatever questions one can raise about
definitions, the notion of definition itself is independent from logic. We can think
about the concepts we need for the construction of a system of logic and construct
a system of logic, without bringing in the notion of definitions at all.
As our deliberations above have shown, what we require for logic is: clear,
precise, well understood terms, without spatio-temporal or pragmatic relativity.
Providing definitions for the terms figuring in a syllogistic inference may be at
times a good means to achieve the above presupposition, but it is not a necessary
means, and at times even if we have definition their availability does not entail
that the presupposition is met.
The myth that there are serious links between logic and definition is likely
to have a historical origin. The Kneales report that Aristotle thought Plato to
be much interested in definitions 22 and that this in turn led Aristotle to think
much about definitions in relation to logic. This historical reconstruction rests on
shaky grounds. Historians tend to confuse two questions: “was Plato interested in
answering ‘what is F’ questions” where ‘F’ stands for what we would call today
an important property such as justice, friendship, insight, etc.? and “was Plato
primarily interested in finding definitions for important qualities, or properties, be
these in the ethical or mathematical realm?” (These were not as separate for him
than for the modern world, for reasons into which we cannot enter here.) In my
case, the answer is affirmative to the first question and negative to the second.
There are many ways to answer a question of the form: “what is it to be a
positive integer? or “what is justice?” For one, a philosopher might give a variety
21 Kneale, VV. and M., Op.cit., p.17. 22).
22 Ibid ., p. 21.
16
Julius Moravcsik
of unique and necessary descriptions of one of these concepts, without being able,
or feel the need for, defining them. There are also many ways of leading an
audience to understand a basic notion without defining it. An obvious example
is ‘language’. There is no general definition for this word, and yet linguists and
philologists manage quite well (making progress in their disciplines, relying on a
common understanding within the profession of what language, and a language
is). Plato never defines ‘number’, and yet reading the relevant texts one comes to
grasp what Plato’s notion of number was. At other times a notion is basic and
primitive, or undefinable in a system, but we come to understand it by considering
its use, and thus how it plays roles in a variety of types of sentences.
The Kneales stress that for Plato it was possible to come up with definitions
that are non-arbitrary and informative. But this shows once more how the ques¬
tions about definitions are independent from those concerning logic. As long as
the presupposition stated above is met, whether the terms used in syllogisms are
arbitrary or not, and whether their connections are trivial or informative does not
affect logic at all. One can construct logics while believing that all definitions are
always arbitrary. Then again, one can do logic with different assumptions. These
issues do not affect the task of proposing rules of valid inference over a domain
and a set of sentences.
What the Kneales might have had in mind, and what should be said, is that
the issue of arbitrariness will affect, not the potential to generate a logic, but the
potential of the definitions used to introduce terms to have explanatory power. But
as we saw earlier, the question of what is a logic and what has explanatory power
are distinct, though in results there can be an overlap. Some explanations can be
phrased in terms of inferences, and some, but not all, can have explanatory power.
E.g., “the larger the state, the more likely a high level of corruption; so-and-so is
a large state; therefore there is a likelihood of high level of corruption.”
Perhaps those insisting on conceptual connections between definitions and logic
have in mind that both involve sketching links between concepts. A definition
typically has the form: “A = B + C”. Thus we “carve up” a concept, and hope
that B and C are sufficiently clear, and thus some illumination is fostered by this
scheme. Likewise, a syllogistic argument reveals certain conceptual relations. But
this is a very thin common denominator. Many other investigations and enter¬
prises also reveal conceptual relations. For example mere classificatory systems,
or drawing contrasts. Furthermore, the ability to provide adequate definitions
does not entail having also the ability to reflect on the nature of definitions and
definition giving. My claim is here, in any case, that neither of these intellectual
exercises has anything significant to do with logic.
This defence of the sudden emergence of logic and its preconditions does not
say that once we have a logic, it cannot be expanded. Surely, that is the way to
construe the relation between syllogistic logic and modern symbolic logic. The lat¬
ter does not replace the former, but provides a larger framework on the same level
of abstractness within which the data syllogistic logic explains can be explained
along with lot more.
Logic Before Aristotle: Development or Birth?
17
A few more reflections on definitions. Some of these may be covering a “closed”
domain, but others do not. Perhaps an account of the series of positive integers
in terms of a starting point and the successor function will generate just the right
class. But consider the definition of “vehicles for transportation”. Given changes
in technology and modes of transportation, there is no way of delineating a sharply
specifiable class as the extension that this term and its meaning gather up. Many
definitions are of this sort. E.g., ‘cooking’. Whatever definition we generate, will
it cover cooking with a microwave oven? And if it does, then why not just any
mixing of edibles, e.g., cereal with yogurt? Currently we think that cooking should
involve heating and some transformation of substance; but modes of heating and
degrees as well as kind of transformation of substance need be left open, given
developments of skill and technology.
In summary, definition is neither a necessary nor sufficient condition for the
kind of vocabulary within which a system of syllogistic logic can be formulated.
Perhaps, the historians’ misjudgments arise from a confusion between what it
takes to have a logic and what it take to have an axiomatic system capturing a
theory, e.g. Euclidean geometry. The second enterprise demands a lot more than
the first. We can reflect on validity and have a logic for a set of sentences without
having an axiomatic system. And as was stressed earlier, having a rigorous way
of dealing with one specific domain, e.g. geometry, does not yet indicate that
the researchers have a conception of logical analysis completely regardless of any
particular domain.
To conclude this section, let us consider some of the things that Aristotle says
about definition, and see if these texts suggest any historical dependence of think¬
ing about logic.
In Post. An. Bk II # 7, Aristotle considers relationships between definition,
demonstration and essence. His conclusion is that one cannot demonstrate essences
by definition. His basic point is that a definition is merely a picture of a configu¬
ration of conceptual relations. Thus it is an articulation, but not a demonstration.
In order for it to be a demonstration it would have to contain inferences. But
a definition is not a series of inferences, even if its ingredients could — in some
other contexts — serve as terms in syllogistic arguments. Furthermore, definitions
cannot prove existence, but deductions with explanatory power must end up with
conclusions about conceptual relations among ingredients that exist. The tone of
this section does not in any way suggest that Aristotle thought of logical infer¬
ences as a result of thinking about definitions. On the contrary, the comments
about definitions seem to be directed to those who did think that there was a
close relationship, perhaps even development. Aristotle’s comparison stresses the
differences, and not the surface similarities.
The discussion continues and in # 10 new light is shed of the relationship
between definitions and syllogistic deductions. In 94a ff. he concentrates on “real
definitions”. Unlike the modern interpreters of the real-nominal distinction, for
Aristotle the key issue is that the real definitions carry existential import. In this
sense we define kinds that exist in nature and not just verbal expressions. He then
18
Julius Moravcsik
explains what definitions are by contrasting these with deductions. The English
“demonstration” is sufficiently polysemous so that one can see what disturbed
Aristotle’s audience. On can think of a demonstration is merely drawing a picture
of conceptual relations, or as a dynamic process that draws out of material initially
understood other material that we did not think was contained in the premisses.
Deductions and inferences are not static pictures. These represent dynamic
processes in which new information comes to light, even though we also learn that
the information was already potentially in the premisses. In this way inferences
are not subject to what was called much later “the paradox of definitions” (“if
the definition is informative, it cannot be right; if it just repeats what we knew
already, it is trivial”.) So in definitions, no inferences; and if no inferences, no logic.
Aristotle seems to have worked out the theory of what a logic is quite independently
of definitions; and in the passages briefly surveyed he contrasts deductions with
definitions, to show how the former are instructive and informative in ways in
which the latter cannot be, though they may be useful for other purposes.
4 LOGIC AND THE METHOD OF DIVISION
The Kneales say that Plato’s Method of Division must have influenced Aristotle’s
thinking about logic. 23 Others too have seen in the past links between the Platonic
method and Aristotle’s logic. We can divide the question into two parts: first, is
there any conceptual link between the Divisions and syllogistic logic, and second,
is there any evidence that Aristotle thought there to be such links? The following
will support negative answers to both of these questions.
Plato’s Method of Division is introduced in Phaedrus 265d-e. In this passage the
Method is introduced both as collecting the right elements under the appropriate
genera, and as the correct way of dividing generic Forms into the “right” species.
In places Plato construes the project as that of coming up with “right namings”.
Clearly any generic Form can be divided into species in an infinite number of ways.
But Plato thinks that some of these are better at reflecting real differences and
similarities between Forms than others. As he says in the Politicus , we must cut
“along the right joints”. 24 There seems to be, then, three tasks that the Method
must fulfill. First, to carve out what in later literature are called “natural” kinds.
Second to present a correct conceptual anatomies of generic Forms so that the more
correct subdivisions will mirror the more important conceptual relations under a
genus, thus vindicating indirectly also the positing of a given genus. Thirdly, by
dividing genera, and then subdividing the results of the first cut, as well as those
that follow, to end up with a series of characteristics, with more and more narrow
extension, all of which can be collected and thus give a “definition” or unique
necessary characterization of a given item under investigation. Examples include
statesmanship and sophistry. (See, e.g., Sophist 258c-d). Since, e.g., sophistry is
23 Ibid., p. 10.
24 Plato, Politicus, 252 d-263b.
Logic Before Aristotle: Development or Birth?
19
illuminated by seven different divisions and none of these is declared as fraudulent,
we can assume that the divisions are not meant to provide one correct account of
a given element, to the exclusion of all others. On the other hand, the fact that
much more space is devoted to one of the characterization of sophistry suggests
that Plato did not regard all divisions to be of equal explanatory value.
If we are to support the Kneales’ conjecture, we would have to argue in the
following way. In at least one of their employment divisions can lead to definitions.
But definitions are linked to logic. Therefore, divisions are linked to logic. In
reply it should be pointed out that, as we saw, no conceptual dependency can be
established between definitions and logic. We can have a fine syllogistic logic as
long as the terms used are clear. For this we need not use definitions, though in
some contexts these can be used to achieve clarity of terms.
Looking closer at what steps we find in divisions and in logical deductions, we
see important differences. Suppose someone wants to “divide” the generic form,
“discipline”, (“art”?) into two sub-species, e.g., productive and exchange-oriented.
How is this cut achieved? There is no way of deducing these two from the genus,
and more importantly, there is no evidence that Plato thought that the cut is the
result of deduction. Rather, the cut is a matter of making the “right” conceptual
divisions. Looking at it one way, repeated cuts along many lines should lead to
appropriate classifications. Once we have these, we can use these as the basis
for constructing — with additional premisses— deductions. But we cannot test
such classifications with deductions. Suppose that we divide humanity into Greeks
and non-Greeks. As long as we did not leave out any element, the division is, in
terms of logical form, correct. Whether it gives us insight is then a quite different
question that mere logical devices will not answer.
Thus Plato’s purpose with his Method is quite different from the purpose of a
syllogistic logic. It is conceptual clarification, with the criteria for success being
given by the notoriously intractable notion of insight or wisdom.
Aristotle’s purpose with syllogistic logic is to present ways of argument that re¬
gardless of subject matter will always enable us to see consequences of conceptual
relations that we assumed at the outset. Logic will show us appropriate infer¬
ence patterns, regardless of the nature of the premisses with which we start. The
premisses may come from any domain, and they may be true or false. Logic is
concerned with inference (topic-neutral), while Plato’s Divisions aim at appropri¬
ate in this or that conceptual domain. (Maybe he saw, by the time he wrote the
Philebus , that in different domains we need different types of divisions. Some func¬
tion along the genus-species line, others in terms of quantitative and measurable
difference.)
So much for one of our two questions. We now ask whether there is any evi¬
dence that Aristotle regarded Plato’s Division as either suggesting logical form, or
functioning as a help in deductive arguments or our checking these.
The key evidence is supposed to be Post An. 96b25 ff. But in this passage
Aristotle makes it quite clear that divisions do not prove or demonstrate anything.
He does want to point out a way in which divisions are nevertheless helpful. But
20
Julius Moravcsik
in the examples he gives all we see is that divisions are useful as a starting point
for forming definitions. They help us in “collecting” all of the attributes that
will explain the essence of the given concept under investigation. As we saw, this
does not show either that there is any conceptual relation between definitions and
logical inferences, or that Aristotle thought so.
Divisions help to put attributes in the right order, i.e., as attributes with less
and less extension, says also that divisions are a great way of checking whether
we got all of the crucial attributes that jointly define the sought essence. But he
should have added that this is so only if we did the division “appropriately”; i.e.
not added anything superfluous and not left out key attributes. But what guara-
tees appropriate dividing? Certainly not anything having to do with deductions.
No logical proof can be given that element E must be a part of the division. This
judgment is based on conceptual intuitions, that lead us to view various elements
as part of essence. Judgments of essentiality may be crucial for certain ways of
viewing classifications, but these intuitions are quite different from logical intu¬
itions of validity, consistency, or inconsistency. It is true that segments of divisions
can be explicated in structures on’ the basis of which deductions can be formu¬
lated. For example, maybe all As are B s, and all B s are Cs. This, so far, has
nothing to do with inferences. Given this conceptual map, one can infer that all As
are Cs. In this context the Method of Division can provide data for deduction, but
lots of other investigations provide data as well. We can draw logical inferences
on the basis of conceptual relations presented in legal presentations. This hardly
shows that Aristotle was influenced in his thinking about logic by law.
In sum, Aristotle’s comments on Divisions can be best interpreted as negative
and defensive. He wants to show how different the purpose and carrying out of
a division is from drawing logical inferences. His remark that divisions can help
in checking proposed definitions are only meant to show that there may be some
intellectual use of these structures. We have evidence that divisions were practised
a great deal in Plato’s Academy. Aristotle says the more or less appreciative things
about divisions, to suggest that all the people who spent so much time and energy
in constructing divisions did not wasted a lifetime.
5 SUMMARY
In comparison with other disciplines, logic moves on a higher level of abstraction.
In physics, for example, there are specific events to account for, then general¬
izations that can stretch into lawlike ones, and after layers of these, a cohesive
system, or theory. In logic we need to analyze and evaluate individual statements,
and such analyses are generalized in terms of consistency and validity. Thus we
have rules of construction and inference. Placed together in a coherent way this
yields a system of logic. We can then reach a higher level of abstraction and look
at systems of logic. Hence theorems about completeness, incompleteness, etc. We
can view the natural sciences as forming theories by talking about entities that
Logic Before Aristotle: Development or Birth?
21
give causal or other types of explanation of sensible particulars. Logic obviously
cannot be viewed this way. Furthermore, logic is not some sort of an abstrac¬
tion from se'nsory experience, nor can it be defined in terms of other disciplines
or interest. Logic has — like mathematics — autonomy and the highest level of
abstractness.
We saw that logic needed a set of terms, vocabulary items, of its own. But we
must not think that once we have some of these terms like predicate, negation,
then we automatically have a logic. To rise to the level of logic we need a special
free creative move of the human mind. Thus logic is not a slow development of
ideas, though once we have logic, it can be expanded, as we saw in the case of
modern symbolic logic.
We need certain linguistic arid thus also, conceptual development in order to
reach the level at which the vocabulary of logic can be forged. Much of this chapter
is devoted to a sketch of how the required vocabulary items can be seen as a last
jump in an otherwise long conceptual and semantic development. The Greeks
reached the required level in one way. Other cultures with other languages might
reach it in different ways.
Our main point is that the rise of logic is both a matter of development and the
matter of instantaneous creation. The required vocabulary must have a historical
process preceding it. Once that is in place, the possibility of constructing a logic
is there. Aristotle was the first to understand the autonomy of logic, and the way
it opens up a magic world of endless explorations of a unique mode of reflection,
construction, justification, and argument evaluation.
APPENDIX 1: DID ARISTOTLE BASE LOGIC ON SOLID FOUNDATIONS?
As we saw, Aristotle’s logic is not only syllogistic logic, but in terms of its inter¬
nal anatomy it is a term logic. The premisses and conclusions contain different
relationships between what one might call today the extensions delineated by the
terms in the arguments. This entailed regarding predication as a key connecting
element between the terms within arguments. Thus Aristotelian logic depends for
its sound foundation on the intelligibility of predication. As long as we construe
the relations between the terms as presented in an argument as overlaps of differ¬
ent sorts between extensions, predication may not seem problematic. The class of
all animals contains the class of humans and that class the class of all Greeks, for
example. Still, Aristotle is concerned with not so much relations between classes
(if one can call the denotations of the terms that) as with the unity of sentences
of subject-predicate form. The copula does not merely relate terms, it is also re¬
sponsible for the unity of the relevant sentences. The sentence that we can assess
as true or false is not merely a collection of parts. It is in some sense more than
the mere sum of parts. And this “more” is indicated by the copula of predication.
Then pushed to the limit, Aristotle retreats to metaphor. In Met. Z, 1041blD-20 he
compares a sentence to a syllable. The syllable is not a mere sum of letters. The
22
Julius Moravcsik
letters in certain juxtapositions yield a syllable, without adding another element
as the connector. It is clear that Aristotle chooses this metaphor so as to avoid
infinite regress. If the copula denotes another element, then there must be a con¬
nection between this element and the predicate, and so on. Is this a satisfactory
answer? Our evaluation depends on what we are willing to take as primitive.
Predication was also a problem earlier, for Plato. But what made predica¬
tion problematic for Plato was different from what puzzled Aristotle. It is worth
mentioning in this context that it is senseless to ask simply: “is predication a
sufficiently clear notion?” This question can be raised meaningfully only within a
conceptual framework. In different frameworks different factors might make pred¬
ication problematic. Is there a framework in which predication can account for
the facts that it must explain in semantic analysis, and still there are no features
of the framework that will render predication puzzling? The jury is still out on
that one.
Plato’s puzzle was different from Aristotle’s because his analyses of sentences of
subject-predicate form were different. There are passages that show without doubt
that in in modern parlance we must represent Platonic predication as tying sub¬
jects to intensional elements. For example, according to Plato everything partakes
of Being, Sameness, and Difference, but there are three distinct Forms involved,
not just one. 25 Thus we cannot take predication as just relating classes. Predica¬
tion, in important contexts., relates spatial particulars to timeless, non-spatial, in
principle recurrent, entities. Hence the special problem of what partaking is, the
relation Plato introduces to make the combining of subject and predicate possible.
Parmenides 131 shows that Plato is concerned with explaining what partaking
is. Not surprisingly, his descriptions are typically negative. It is not part-whole
relations, it is not physical engulfing, and so on. The only positive account turns
out, not surprisingly, to be metaphoric. In the Sophist he introduces the notion
of a Vowel-Form, and regards Being (in the predicative sense) as a prime exam¬
ple. Thus Being has as its sole function to relate things; it functions like vowels
connecting consonants.
So much — briefly — for ancient wisdom on the topic.. Do we do better when
looking at modern proposals? The most profound statement of the worry can be
found in the writings of Frege. He compares the subject-predicate form to that
of function and argument. Functions are incomplete, unsaturated, as Frege says,
and thus their joining with arguments leads to intelligible completeness. Is this a
better solution than the earlier ones? Is Frege’s metaphor better than the earlier
ones by Plato and Aristotle? Arguments still rage on that issue. 26 In his masterly
review of twentieth-century proposals Frank Ramsey adds Wittgenstein’s proposal
according to which objects in atomic propositions hang in one another like the links
of a chain. 27
Ramsey himself attempts to draw the distinction by viewing the necessity of the
25 Plato, Sophist 255b-d.d
26 Essays on Frege, 1968, ed. I. Klemke, (Chicago: University of Illinois Press.)
27 Ramsey, F. 1931, “Universals”. in The Foundations of Mathematics, pp. 112-135; p. 129.
Logic Before Aristotle: Development or Birth?
23
subject-predicate for as arising only in atomic propositions, and he construes these
as having an ontological structure of a property being predicated (or universal) of a
spatio-temporal individual. Thus the issue in Ramsey’s analysis boils down to this:
Is there a non-question-begging way of distinguishing universals from particulars?
Borrowing an idea from Whitehead, Ramsey sketches in a few pages how one can
consider a particular “adjectivally” e.g., regard the particular object as adjectival
to events Needless to say, we can continue this by moving from one type of events
to others, and so on.
One could continue and enumerate more recent proposals such as those by
Strawson and others, but these are basically variations on the same theme and
Ramsey’s application of the Whiteheadian idea remains intact.
Why, then, should we look at predication in this puzzled way? The origin of
these puzzles come from the monism of Parmenides who declared predication to
be incoherent. He claimed that statement of subject-predicate form entail also
negative statements. To say that x has F, makes sense only if we add to it that
in virtue of this it is also something negative, a not-G for example. Parmenides
questions the intelligibility of not being G. Plato responded by saying that in the
contrast of what is fine and what is not fine two classes (collections?) of existent
entities are juxtaposed. This hardly answers Parmenides who would question the
ontological status of the negativity of the not-fine, “not-G”, etc.
One might wave one’s hand at other modern treatments of predication. It, and
with it class membership, or instantiation, are basic primitives, we need then in our
pluralistic analysis of reality, and as long as employment of predication does not
lead to formal contradictions it is legitimate to use it. Whether such a complete
separation of what is to be regarded as legitimate in logic and what are viable
ontologies underlying the use of logic and language is intellectually conscionable
or not cannot be discussed here. The fact is that after 2400 years all philosophers
have come up with as response to the Parmenidean challenge is a bouquet of four
metaphors. If there is immortality, then Parmenides is right now chortling in his
coffin.
APPENDIX 2. LOGIC AND GRAMMAR: EPISTEME, TECHNE,
EMPEIRIA?
We can learn about the status of logic by a comparison with grammar, and es¬
pecially the discussions about grammar in the early stages of its establishment in
the Greek scholarly world. 28
Was grammar to be regarded as an episteme (branch of knowledge) or techne
(rational discipline) or ere empeiria (set of empirical conjectures)? The difficulty
surrounding these debates is that these three terms changed their meaning over
28 Steinthal, H. 1891, Geschichte der Sparchwissenschajten bei den Griechen und Roemern,
Vol. II. (repr. Hildescheim, Germany), pp. 173-178.
24
Julius Moravcsik
time. For Plato in the Gorgias the issue was, given a certain putative discipline, is
it genuine knowledge or mere empirical beliefs? This question cannot be applied
to grammar in a straightforward way, because the nature and scope of grammar
was construed by different authors in different ways. On the one extreme one can
view grammar as a descriptive study that is to record all of the language uses by
a linguistic community. In this way of looking at it we come up with descriptions
many of which are not even lawlike. Furthermore, many of the rules of grammar
and the formulae allowed strike one as arbitrary.
At the other extreme one can think of grammar as a normative discipline. It
does not describe how we do speak, but how we ought to speak. The question
emerging at this stage is: what is the authority underlying the “ought”? Some will
say the speech of the poets, philosophers, and scientists. Others might accept this
but add in any case that at the base of grammar we find certain necessary features
that enable grammar to reflect some of the basic patterns of reality. Needless to
say, there are many possible positions in between.
Reflecting the ambivalence is the use of the phrase “epi to polu”. For a staunch
empiricist this means “the usual’ or “for the most part” where these notions are
spelled out probabilistically. For Plato and Aristotle this was not the appropriate
meaning. For Aristotle the phrase represented what in modern English we call the
generic use. An animal does something “epi to polu ” if in doing so it expresses
an aspect of its essence; e.g., “beavers build dams”. This means, roughly, “the
normal, healthy beaver.” It need not be the majority of beavers (maybe many are
sick), or the statistically predictable ones. Correspondingly if you want a grammar
to reflect how people speak “epi to polu ”, you want to write a purely descriptive
grammar, or one that tells people what linguistic use in the case of this language
is at its best.
Interestingly, the more sparse the grammar, the more plausible the normative
interpretation seems. Thus Plato in Sophist 258-259 introduces a complex that is
roughly equivalent to actor/agent vs. action/property as fundamental to sentences
expressing truth or falsity, and thinks that this reflects the metaphysical relations
between Forms. Thus the structure posited is both necessary and justified. Need¬
less to say, one cannot call this a complete grammar. Some such structure remains
necessary and normative in Aristotle’s treatment of basic combinations. But as
Aristotle adds to the “grammar” e.g., endings, the ground for regarding his basic
structures as based on features of reality seem weaker, and with that, of course,
also the normativity.
How does all of this apply to logic? In principle one could start an enterprise of
discovering and describing how people in fact reason. This is clearly not Aristotle’s
enterprise. The system of logic Aristotle articulates is not a piece of psychology.
It shows us how we ought to reason, should reason, and not what we in fact do,
though one might say that in Aristotle the “ought” in this context translates into
“how humans at their best reason”. His teleological conception of reality endowes
logic at its base with both necessity and normativity. It is interesting thus to note
that the necessity and normativity of certain combinations of linguistic units have
Logic Before Aristotle: Development or Birth?
25
as their origin Plato’s reflection on minimal grammar-like combinations which, in
turn rest on their alleged isomorphism with some fundamental ontological relations
among the most basic constituents of reality, namely the Forms. Denuded of the
Platonistic metaphysics, but keeping some form of the necessary in reality and
its reflection on language yields these two essential aspects of logic necessity and
normativity.
This Page Intentionally Left Blank
ARISTOTLE’S EARLY LOGIC
John Woods and Andrew Irvine
1 BIOGRAPHICAL BACKGROUND
Aristotle is generally recognized as the founder of systematic logic, or of what
he called “analytics.” For over two thousand years he was logic’s most influential
writer.
Although there were precursors, especially with respect to the study of dialectic
or the art of public argument, Aristotle was the first to systematize universally
valid logical laws. As Julius Moravcsik points out, “[we do not] credit a logician
with originality when he first presents the law of contradiction. 1 But we do credit
him with originality in view of the particular ways he shapes rules for deduction
and proofs.” 2
Aristotle was also responsible for the remarkable accomplishment of developing
logic in at least two distinct ways, including his almost complete theory of the
syllogism and his complex and sophisticated theory of modal logic. In addition,
he is noted for his work in axiomatics, and there is some evidence that he also
began investigating what is now called propositional logic, although he did not de¬
velop these investigations systematically. His claim near the end of On Sophistical
Refutations that he is the primary creator of the discipline of logic is therefore
quite justified (On Sophistical Refutations , 15, 174°, 20).
Aristotle was born in 384 BCE, in Stagira in northern Greece, and it is from
here that he received his nickname, “the Stagirite.” His father, Nicomachus, had
been court physician to the Macedonian King, Aourntas II, but both his parents
died when Aristotle was still a boy. In 367 BCE, Aristotle was sent to Athens
to study under Plato. He remained at Plato’s school, the Academy, first as a
student, and later as an instructor, for almost twenty years. Initially, the students
at the Academy made fun of this new foreigner who spoke with a lisp, but Plato
himself was impressed and nicknamed Aristotle “the intelligence” of the school.
Upon Plato’s death, Aristotle left Athens, perhaps for political reasons, or perhaps
because he had not been appointed Plato’s successor and was dissatisfied with how
the Academy was now being directed. After travelling for awhile he married the
(adopted) daughter of a former classmate who had become an Aegean king and in
whose court Aristotle for a time served.
1 Cf. Plato at Republic 436B: “It is obvious that the same thing will never do or suffer opposites
in the same respect in relation t.o the same thing at the same time.”
2 Moravcsik, this volume, ch. 1.
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
@ 2004 Elsevier BV. All rights reserved.
28
John Woods and Andrew Irvine
In 342 BCE Aristotle was appointed by Philip II of Macedon to tutor his son,
the future Alexander the Great. From Aristotle’s point of view, the appointment
could not have been a pleasant one. He was expected to provide guidance to the
young Alexander in a palace noted for its savagery and debaucheries, and at a time
when both the father and son were intent upon assassination and conquest. When
Alexander finally claimed his father’s throne in 336 BCE, Aristotle left the court.
Some reports say that Aristotle left well endowed by Alexander; others say that he
was lucky to escape with his life. In any event, his influence could not have been
too great since Alexander soon ordered Aristotle’s nephew, Callisthenes, hanged
for refusing to bow before the new king.
When Aristotle finally returned to Athens in 335 BCE, he opened his own
school, the Lyceum, in a grove just northeast of the city. At the Lyceum he
planted a botanical garden, began both a library and a natural history museum,
and taught philosophy. The school was also known as the Peripatos or “strolling
school” since, at least occasionally, Aristotle lectured to his students while strolling
about the school’s grounds. It is for this reason that even today Aristotle’s follow¬
ers are known as “peripatetics.” In 323 BCE, following the death of Alexander,
anti-Macedonian feeling in Athens increased and, because of Aristotle’s ties to
Alexander’s court, the accusation of impiety was raised. Recalling the fate of
Socrates—who had been put to death after being found guilty of similar charges—
Aristotle returned to Chalcis, his mother’s hometown, saying that he did not want
Athens to “sin twice against philosophy.” He died the following year, in 322 BCE.
Aristotle’s interests were universal. His writings represent an encyclopedic ac¬
count of the scientific and philosophical knowledge of his time, much of which
originated with Aristotle himself and with his school. His extant writings cover
logic, rhetoric, linguistics, the physical sciences (including biology, zoology, astron¬
omy and physics), psychology, natural history, metaphysics, aesthetics, ethics and
politics.
His logical writings, referred to as the Organon , consist of six books; the Cate¬
gories ., De Interpretatione (On Interpretation ), the Prior Analytics, the Posterior
Analytics, the Topics, and De Sophisticis Elenchis (On Sophistical Refutations).
Transmitted from the ancient world in large part through Arab scholars, Aristo¬
tle’s writings shaped the intellectual development of medieval Europe. His logical
treatises occupied a central place within the medieval curriculum and it was during
this time that Aristotle came to be known as the source of all knowledge. Even
so, the first modern critical edition of Aristotle’s writings did not appear until
1831. The scholarly practice of citing Aristotle’s work by a series of numbers and
letters still refers to the page, column and line numbers of this nineteenth-century
edition.
While it has often been claimed that Aristotle’s dominating authority over such
a long period hampered the development of logic (just as it did the development
of science), such an observation cannot properly be taken as criticism of Aristotle
himself. Instead, it is a telling incrimination of those of less talent and imagination
who were to follow his remarkable accomplishments.
Aristotle’s Early Logic
29
2 MOTIVATION
The Greek intellectual revolution can be characterized in large part by its dis¬
covery of a new method of enquiry and demonstration. This new method, called
logos , shared its name with a perceived rational purpose thought to underlie the
entire universe. Thus in one sense, logos represented the laws and regularities gov¬
erning all of nature. In another, it represented the process of reasoning by which
these laws and regularities were to be discovered. This new method of reasoning
originated in physics and, with cosmology as a bridge, soon began to influence
all branches of knowledge. Eventually, however, it was to collapse into a kind of
intellectual pathology, typified by the efforts of pre-Socratic philosophers such as
Heraclitus and Parmenides. Pathological philosophy was logos run amok and, for
all its quirky theoretical charm, logos was soon being regarded as an intellectual
disgrace. Left to its own devises, it threatened to destroy science and common
sense alike.
It is widely agreed that the destructive arguments of Heraclitus and
Parmenides, as well as those of many of the most able of the Sophists such as
Protagoras and Gorgias, turn on the mismanagement of ambiguity. In the case of
Heraclitus, his repeated equivocations typically take the following form:
If v is $ in one sense and not-# in another sense, then v is both $ and
not-#.
On the other hand, Parmenides’ equivocations are typically driven by a mis¬
conception which is the dual of Heraclitus’ error:
If v is $ in one sense and not-# in another sense, then v is neither #
nor not #. 3
Since Heraclitus and Parmenides both appear to accept a common major pre¬
miss, namely that
For all v and for all #, v is $ in one sense if and only if v is not-# in
another sense.
there arose the two great pathological metaphysics of the ancient world. For Her¬
aclitus, the world turns out to be thoroughly inconsistent (or, as modern logicians
would say, absolutely inconsistent) while for Parmenides the world turns out to be
thoroughly indeterminate (or, as modern logicians would say, non-truth-valued).
No doubt it will strike the modern reader as puzzling that these blunders were
accorded such high respect by thinkers as able as Plato and Aristotle. How could
anyone be fooled by such blatant equivocations? In answer, there are two pos¬
sibilities. One is that the predecessors of Plato and Aristotle could not see the
3 Lawrence Powers calls these, respectively, the Heraclitean Rule and the Parmenidean Rule
[Powers, 1995, ch. 2].
30
John Woods and Andrew Irvine
equivocations that they were guilty of. The other is that they indeed had the
intuition that they somehow had mismanaged ambiguity, but that for some reason
these intuitions appeared to be untrustworthy.
Like Plato, Aristotle wrote copiously, and he is at the very height of his intellec¬
tual powers in his discussions of these types of pathological philosophy. According
to Aristotle, logos had been used in ways that denied both the reality of the empir¬
ical world and the meaningfulness of language. Aristotle saw Plato’s forms as an
important attempt to de-pathologize philosophy but, largely for reasons set out in
the Parmenides , he also sees Plato’s project as a serious failure. Aristotle is thus
left with the following fundamental question: How can logos be made to behave ?
Aristotle’s answer was to invent logic, and to use logic as a constraint upon
logos. Logic would be a technique, or a set of techniques, for facilitating the use
of correct reason, for constraining and taming logos.
It is interesting that the first two monographs of the Organon, the
Categories and On Interpretation, seem not to be about logic at all, never mind
their occasional references to logical concepts and principles. The Categories con¬
tains an elaborate taxonomy of types of change, and On Interpretation is a theory
of grammar for the Greek language. In both these works, Aristotle also devotes
considerable attention to the phenomenon of ambiguity and to the deductive cor¬
ruptions to which it gives rise; but when it is borne in mind that Aristotle is
taking up the challenge of de-pathologizing philosophy, and that he seeks to do
this by holding all theoretical reasoning to the standards of a correct logic, it is
not surprising that he should start with an examination of change and ambigu¬
ity. Heraclitus and Parmenides both emphasize arguments that exploit change
and ambiguity, and Aristotle is of the view that such arguments turn pivotally
on errors in the ways in which change and ambiguity are to be analyzed. As a
result, he begins his great reform of philosophy with an attempt to conceptual¬
ize these matters correctly. Plato had been shrewd enough to see that some of
these pathological arguments go wrong because of equivocation—the conceptual
mismanagement of ambiguity; but Plato’s own attempts at repair reveal that he
lacks a competent understanding of equivocation ( Republic 479B ff.). So, again,
it is wholly tempting to see in the works of both Plato and Aristotle a response
to the Heraclitean-Parmenidean challenge. This is especially true of Aristotle. In
these first two books he is struggling to produce a theory of ambiguity and a set
of protocols for its avoidance.
Following the Categories and On Interpretation come two works of signal
importance—the Topics and On Sophistical Refutations. These two monographs
are closely connected; in fact, some scholars are of the view that On Sophistical
Refutations is either a ninth chapter of, or an appendix to, the Topics. By “topic,”
Aristotle means a “strategy” or “scheme of argument”; so we must not confuse
this word with our word, which means “subject-matter.” The importance of these
books consists primarily in Aristotle’s insight that there exists a model of correct
argument which has a wholly general application. This model of argument is the
Aristotle’s Early Logic
31
syllogism, , and with it comes a precise answer to Aristotle’s question of how logos
can be made to behave.
Showing that all correct reasoning—all legitimate use of logos —conforms to
Aristotle’s theory of the syllogism would be a stunning accomplishment. It would
establish that errors in reasoning arise solely for reasons other than the use of
topic-neutral models of argument, to recur to our use of the word “topic.” In this,
Aristotle stands apart from more familiar Socratic denunciations of the teacher-
generalist. He proposes to make good on the Sophist’s central insight, that there
are correct modes of reasoning and correct ways of arguing about anything and
everything.
As we proceed, there will be several ways in which we will be able to judge the
theoretical fruitfulness of Aristotle’s invention of the syllogism. That logic was
invented by a philosopher is a significant fact. Many a profession could claim the
indispensability of clear thinking for sound practice. So why was logic not invented
by an admiral or a general, or by a physician or a physicist? Why indeed was logic
not invented by a mathematician: why is Aristotle not the Frege of the ancient
world?
Logos is nothing if not a corrective to common sense. Logos has an inherent obli¬
gation to surprise. It began with the brilliant speculations of the Pythagoreans—
the original neopythagoreans, as one wag has put it—with regard to a number-
theoretic ontology. Apart from the physicists, the great majority of influential
practitioners of logos before Plato allowed logos to operate at two removes from
common sense. The first was the remove at which speculative science itself would
achieve a degree of theoretical maturity. But the second remove was from sci¬
ence itself. The first philosophers were unique among the practitioners of logos in
that they created a crisis for logos. In the hands of the sophists, philosophy had
become its own unique problem. It was unable to contain the unbridled argumen¬
tative and discursive fire-power of logos. In fact, philosophy has had this same sort
of problem—the problem of trying to salvage itself from its excesses—off and on
ever since. Thus, logic was invented by a philosopher because it was a philosopher
who knew best the pathological problematic that philosophy had itself created.
3 ORIGINS
It is widely accepted that Aristotle’s main contributions to logic begin—and some
would say end—with the Prior Analytics. 4 We are of a different view. It is
often remarked that Aristotle may have composed the Organon in the following
order: the Categories ; On Interpretation-, Topics I VII] Posterior Analytics /;
Topics VIII and On Sophistical Refutations-, and Prior Analytics and Posterior
4 Thus it is with scarcely an exception that many of the leading contemporary commentaries
concentrate on the Prior Analytics. In these writings one finds little to suggest that Aristotle’s
earlier treatises might warrant detailed critical scrutiny, even as a fledgling venture into logical
theory. See, for example, [Lukasiewicz, 1957]; [Kneale and Kneale, 1962]; [Patzig, 1968]; [Smiley,
1973]; [Corcoran, 1974a]; [Kapp, 1975]; [Lear, 1980]; [Thom, 1981]; and [Frede, 1987].
32
John Woods and Andrew Irvine
Analytics II. 5 Yet if this is so, a certain caution is called for. 6 7 If, for example,
Posterior Analytics I does indeed precede not only Topics VIII and On Sophistical
Refutations but the Prior Analytics as well, it cannot strictly be true that logic
originates with the Prior Analytics , since book one of the Posterior Analytics
represents a considerable anticipation of many of the formal structures contained
within the Prior Analytics7 Alternatively, on the chronological ranking of Barnes,
in which the Topics and On Sophistical Refutations precede the two Analytics,
themselves written in fits and starts over a more or less unified later period, 8
much the same point can be made. Thi si so, even though it remains the dominant
contemporary view that the Topics and On Sophistical Refutations are treatises on
dialectic and that, being so, they are not a serious contribution to logical theory. 9
Concerning On Sophistical Refutations, Hintikka proposes that
[i]nstead of being mistaken inference-types, the traditional fallacies
were mistakes or breaches in the knowledge-seeking questioning games
which were practised in Plato’s Academy and later in Aristotle’s Lyceum.
Accordingly, they must not be studied by reference to codifications
of deductive logic, inductive logic, or informal logic, for these are all
usually thought of as codifications of inferences [Hintikka, 1987, pp.
211-238]. 10
5 [Forster and Furley, 1955, p. 4], See also [Kneale and Kneale, 1962, pp. 23-24]; [Rist, 1989,
pp. 76-82] and [Dorion, 1995, pp. 25-27].
6 We note, in passing, an interesting contention between Solmsen and Ross. Solmsen holds
the chronological claim in the form of a priority of dialectic ( Topics ) and apodeictic ( Posterior
Analytics ) over syllogistic ( Prior Analytics ). This is stoutly resisted by Ross. But contra Ross,
see [Barnes, 1981]. ( Cf. [Solmsen, 1929]; [Ross, 1949]; and [Forster and Furley, 1955].)
7 For example, [Boger, 1998a] holds that Aristotle’s work on the fallacies in On Sophistical
Refutations presupposes the mature theory of the Prior Analytics [cf. [Boger, 1998b]).
8 [Barnes, 1993, p. xv]. It is, however, a mistake to attribute a full-blown chronology to Barnes:
“Here and there ... we can indeed make chronological claims which have a certain plausibility
to them; and some of these claims are not without philosophical significance. (For example,
we believe that the core of the theory of demonstration which is expounded in An. Post was
developed before the polished theory of syllogistic which is expounded in An. Pr ; and we believe
that this has some bearing on the way we should interpret some of Aristotle’s views about the
nature of science). But claims of this sort will rarely be made with any confidence; they cannot
yield a chronology of Aristotle’s writings; and they will not amount of anything which we could
call an intellectual biography” [Barnes, 1995, pp. 21-22].
9 For example, see Corcoran: “Aristotle presents [his logical] theory almost completed,
in chapters 1, 2, 4, 5 and 6 of the first book of Prior Analytics, though it presup¬
poses certain developments in previous works—especially the following two: first, a the¬
ory of form and meaning of propositions having an essential component in Categories
[Corcoran, 1974b, ch. 5, esp. pp. 234-267]; second, a doctrine of opposition (contra¬
diction) more fully explained in Interpretations (chapter 7). Bochenski has called this the¬
ory [of book one of the Prior Analytics ] ‘Aristotle’s second logic,’ because it was evidently
developed after the relatively immature logic of Topics and On Sophistical Refutations ...”
[Corcoran, 1974a, p. 88], (emphasis added). Modus ponens and modus tollens are recognized at
Topics 111 6 , 17-13 and 112", 16-13, and opposition and negation are discussed at Topics I43 b ,
15 ff.
10 Cf. [Woods and Hansen, 1997],
Aristotle’s Early Logic
33
Hence,
in a sense all Aristotelian fallacies are essentially mistakes in question¬
ing games, while some of them are accidentally mistakes in deductive
... reasoning [Hintikka, 1987, p. 213] (emphasis added). 11
Yet why should this be so? Hintikka believes that it is because On Sophistical
Refutations is a dialectical work, not a logical one. As against this, Hamblin
attributes to Aristotle the view that
[dialectic as a mere technique [is] unessential to the pursuit of truth. 12
At times ... [Aristotle] even thinks of it as a hindrance: he is in the
process of discovering Logic which, he thinks, enables a man to achieve
as much by solitary thought as in social intercourse [Hamblin, 1970, p.
60],
Hence,
Aristotle’s On Sophistical Refutations can then be regarded as a first
step in constructing the relevant logical theory [Hamblin, 1970, p. 59].
Hamblin himself thinks this is a retrograde step, at least when it comes to
handling the fallacies:
[I]n our attempt to understand Aristotle’s account of fallacies we need
to give up our [and Aristotle’s] tendency to see them as purely logical
and see them instead as moves in the presentation of a contentious
argument by one person to another [Hamblin, 1970, pp. 65-66].
In contrast, we find this tension between dialectic and logic somewhat miscon¬
ceived. Even apart from Aristotle’s work on the fallacies, Hamblin is certainly
right to say that Aristotle’s dialectical works mark the beginning of logical the¬
ory. On Sophistical Refutations has as its primary target an analysis of sophistical
refutations with which an unrepentant tradition (mis)identifies Aristotle’s list of
thirteen fallacies. 13 Whatever we may think of this identification, a theory of
sophistical refutation requires a theory of refutation. Aristotle obliges with an
obscure definition:
11 Cf. [Woods and Hansen, 1997, pp. 217-239],
12 This, en passant, is nothing that Aristotle would have accepted. Dialectic is an indispensable
instrument of negative knowledge, of the discovery of what is not the truth, itself essential to the
pursuit of truth. What is true is that Aristotelian dialectic cannot demonstrate positive truth
with certainty. On the other hand, dialectic is also a kind of induction ( epagoge ), a method of
examining all sides of an issue in ways that sometimes gets inquirers to see the self-evidence of
first principles.
13 For helpful discouragement, see [Hansen, 1992],
34
John Woods and Andrew Irvine
For to refute is to contradict one and the same attribute—not the name,
but the object and one that is not synonymous but the same—and to
confute it from the proposition granted, necessarily without including
in the reckoning the original point to be proved, in the same respect
and relation and manner and time in which it was asserted ... Some
people, however, omit one of the said conditions and give a merely
apparent refutation ... (On Sophistical Refutations , 167“, 23-29).
There follows an account of refutations which presupposes an account of syl¬
logisms, and which eventuates in a formal codification, together with something
tantamount to a completeness proof in the Prior Analytics . 14 What is distinctive
about the Prior Analytics is thus not the doctrine of the syllogism; rather, it is,
as we would say today, the discovery of a proof procedure for completeness [Lear,
1980, ch. 2, pp. 15-33]. In other words, the project of the Prior Analytics is
formal and reductive. Aristotle gives two treatments of the syllogism. In the one,
syllogisms are considered informally, and generally. In the other, they are sub¬
jected to formal constraints by which they are fitted for the particular theoretical
purposes of the Prior Analytics. 15
The question of the chronology of the Organon is thus part of the larger issue
of the extent to which Aristotle’s philosophy underwent significant development.
On the developmental view, Aristotle recognized that certain of his later doctrines
contradict and displace earlier doctrines. 16 The developmental perspective is re¬
sisted by some scholars, who hold that, by the time they were complete, Aristotle
saw his writings as forming a consistent and unified whole. 17 Still others hold that
the developmental and anti-developmental perspectives are reconcilable. 18
For our purposes it is unnecessary to decide these larger questions. We shall say
that the logical theory of the Prior Analytics presupposes Aristotle’s theory of syl¬
logisms (and—tacitly—a theory of validity, too). These theories appear implicitly
in the Topics and On Sophistical Refutations. (In passing, we note that to say that
a theory T presupposes a theory T*, is to say that T could not be true without T*
14 AristotIe’s own attempt, which does not quite succeed, is to be found at Prior
Analytics 23. However, [Corcoran, 1972] has shown how to repair Aristotle’s proof.
16 “Indeed there is an ambiguity in Aristotle’s use of the word ‘syllogism’ similar to that in
the modern use of the word ‘deduction.’ There is first the use of ‘syllogism’ in the broad sense
... This corresponds to our use of ‘deduction’ in the general sense of an informal argument in
which the conclusion is a logical consequence of the premisses ... Second, there is the use of
‘syllogism’ in the narrow sense, used to describe the formal inferences and chains of inferences
that Aristotle isolated [in the Prior Analytics ]” [Lear, 1980, p. 10]. Cf. [Mignucci, 1991, p. 25]:
“[T]he definition of ‘syllogism’ at the beginning of the Prior Analytics (24 6 , 18-22) refers to the
generic meaning of the word, and it does not apply to the special inferences of which Aristotle
offers the theory in the following chapter.” It appears that the distinction between syllogisms
in the broad and narrow sense was not recognized by mediaeval commentators. Thus, “[T]he
mediaevals never doubt that [Aristotle] means [by ‘syllogism’] the same in the Topics as in the
Prior Analytics ...” [Green-Pederson, 1984, p. 20].
18 See here [Jaeger, 1923]; English translation in [Robinson, 1948].
17 See, for example, [Cherniss, 1935].
18 See [Graham, 1987], See also [Scott, 1971] cited in [van Benthem, 1994, p. 133].
Aristotle’s Early Logic
35
being true, whereas T* could be true without T being true. The relationship is
preserved even when T* is repeated in T, as is, to a large extent, the logical theory
of the Topics and On Sophistical Refutations in the Prior Analytics.) It is well to
note that the presupposition claim carries no implication of temporal priority.
The Topics is a handbook of dialectical argument that presupposes a distinction
between dialectical propositions and dialectical problems. They differ in three
ways: by way of content, logical form, and function. As to content, “a dialectical
proposition [or premiss] consists in asking something that is reputable to all men
or to most men or to the wise” ( Topics 104“, 9-10). Dialectical propositions are
thus those that are believed to have a prima facie degree of credibility because
they are universally or widely held, or because they are held by someone whose
opinion deserves respect. The Greek term for such a proposition is endoxon. In
contrast, a dialectical problem
is a subject of inquiry that contributes either to choice and avoidance,
or to truth and knowledge, and does that either by itself, or as a help
to the solution of some other such problem. It must, moreover, be
something on which either people hold no opinion either way, or most
people hold a contrary opinion to the wise, or the wise to most people,
or each of them among themselves. ( Topics 104 6 , 1-6).
Dialectical problems differ in content from dialectical propositions in that what
marks them as problems is that their status is unsettled. (We note in passing that,
being questions, so-called dialectical propositions are not a type of proposition in
what we are calling Aristotle’s technical sense.)General or expert opinion is not
clear on what the answer to the problem is. Dialectical problems lack the very
thing that makes a proposition an endoxon.
The logical form of a dialectical proposition is, “Is A BV; for example, “Is
two-footed terrestrial animal the definition of man?” ( Topics 101 b , 29-30). All
dialectical propositions have to have this form, and they must be answerable by
Yes or No ( Topics 158“, 16-17). In contrast, the logical form of a dialectical
problem is that of a disjunctive proposition, “Is A B or is A not -BT'\ for example,
“Is two-footed terrestrial animal the definition of man or not?” ( Topics 101 6 , 32-
33). A question of this form cannot be answered (non-vacuously) by a simple Yes
or No. The answerer must choose one of the two disjuncts, thereby committing
to one of two propositions, either “A is B" or “A is not -B.” We can see that the
logical forms of both dialectical propositions and problems determine the logical
forms of the answers to be given.
We can also distinguish dialectical problems and propositions by their different
functions in refutations. The function of a dialectical problem is to give rise to
a dialectical discussion; in opting for one of the two possible answers, a thesis
is established that will be the target of a refutation. The function of dialectical
propositions, to be answered by a Yes or a No, is to provide the grounds for the
possible refutation of the thesis by being the premises from which the refutation
is fashioned.
36
John Woods and Andrew Irvine
Aristotle contrasts dialectical arguments with other kinds of arguments in at
least two places. In On Sophistical Refutations (2, 165“, 37-165 6 , 12) he lists four
kinds of arguments used in discussion: (1) scientific arguments which reason from
first principles appropriate to a subject and not from opinions of the answerer;
(2) dialectical arguments which reason from generally accepted opinions to a con¬
tradiction; (3) examination arguments which reason from opinions held by the
answerer; and (4) contentious arguments which reason from, or seem to reason
from, opinions which are, or appear to be, generally accepted.
The object is then
to discover some faculty of reasoning about any theme put before us
from the most reputable premisses that are [ endoxa} ... we therefore
proposed for our tretise not only the aforesaid aim of being able to
exact an account of any view, but also the aim of ensuring that in
defending an argument we shall defend our thesis in the same manner
by means of views as reputable as possible. (On Sophistical Refutations
183“, 37-183 6 , 6).
In places Aristotle seems to advance something stronger. His strategies enable
a reasoner to reason about anything whatever, independent of its subject matter
(On Sophistical Refutations 170“, 38; 171 b , 6-7). 19 In other places still, he appears
to confine himself to arguments from definitions and, thus, to arguments that are
not entirely topic-neutral (Topics 102 6 , 27; 120 6 , 10 ff.). This does not cancel
the claim that the arguments under review are always those that reason about
reputable opinions. For it is possible that anything about which it is possible to
argue independently of its content is a possible object of opinion by experts or
by the many or by the wise, but there is nothing in the Topics or On Sophistical
Refutations to suggest that the strategies advanced there have application to such
arguments only on the assumption of some such possibility as this. For example,
Aristotle explicitly recognizes demonstrative arguments, i.e., arguments from pre¬
misses that are true, primary, appropriate to their subject matter and better
known, or more intelligible, than the conclusions that they sanction ( Topics , 141“,
29; 158“, 36-37; On Sophistical Refutations 165 6 , 1 and 172°, 19). Although it is
true that Aristotle contrasts demonstrative arguments with dialectical arguments,
it does not follow that strategies for the engagement of dialectical arguments have
no application to demonstrations.
On Sophistical Refutations concerns itself with various ways in which the ends
of argument can be subverted, sometimes deliberately, and with strategies for
avoiding and evading these pitfalls. The example of refutation dominates this
work. Aristotle specifies thirteen respects in which a refutation can go wrong,
ways in which the purported refutation is not really a refutation but only appears
to be one. These are his sophistical refutations, and here the word “sophistical”
19 Cf. [Allen, 1995]: “But dialectic, the faculty of arguing about all matters, remains possible,
for it falls to the dialectician to know the refutation arising through topoi, which are common
by bearing all subject matters (On Sophistical Refutations 170°, 34-170 6 , 1).”
Aristotle’s Early Logic
37
carries the meaning of “sham” or “counterfeit.” Aristotle’s list of thirteen is famous
to this day. Traditionally, his sophistical refutations have been divided into two
categories, which Latin translators have labelled in dictione and extra dictionem.
In the first are equivocation, amphiboly, combination of words, division of words,
accent and forms of expression. The other category comprises accident, secundum
quid, ignoratio elenchi, consequent, non-cause as cause, begging the question and
many questions. It is customary for commentators to think of the in dictione cases
as sophistical refutations that “depend on language,” and of the extra dictionem
as “depending on considerations other than linguistic ones.” However, this may
not be the distinction that Aristotle intends.
Every putative refutation is an argument of a certain kind, in a sense of ar¬
gument which, for Aristotle, is always a linguistic entity. So it may be said that
any argument, good or bad, owes its goodness or badness to linguistic factors.
Thus, what Aristotle has in mind is not a distinction between linguistic and non-
linguistic considerations, but rather a distinction that turns on whether or not an
argument is spoken.
For example, an argument is brought down by the sophism of accent when it
contains a word which, when pronounced one way has one meaning, and when
pronounced another way has a second meaning. If the argument in question were
written down, the ambiguous word might not reveal its intended meaning, since
it would have only one spelling. But if the argument were spoken, the word would
be disambiguated by its different pronunciations. This is a matter in dictione
precisely because a problem that might cripple the argument in written form would
be cleared up in speaking it. 20 So, whether or not it will bear close scrutiny in
every case, Aristotle’s intention is to capture a class of argument mistakes that
could be avoided by speaking the arguments in question. In contrast, the category
extra dictionem would be made up of argument mistakes that cannot be be spotted
or avoided in this same way, as for example with begging the question.
It is apparent that not everything falling under any of the thirteen subcategories
in Aristotle’s two lists is a sophistical refutation. A sophistical refutation is an
argument which appears to be a refutation but, in fact, is not. Many arguments
that do not even pretend to be refutations are arguments that are made bad by
their instantiation of one or another of the structures in Aristotle’s list. Any such
argument is a bad argument and Aristotle thinks that it is made bad, and often also
is made to appear good, by its instantiating one of the thirteen conditions. When
this happens the argument in question is a paralogismos or a fallacy, and whereas
it is Aristotle’s view that a sophistical refutation is always a fallacy, it is not his
view, nor is it true, that a fallacy is always a sophistical refutation. As a result,
we see here, too, that the theoretical apparatus of On Sophistical Refutations has
an application that extends beyond the kinds of argument denoted by the title of
that work.
20 A charming example of accent is given by Powers: “The workers were unionized and therefore
contained no extra electrons,” which exploits the fact that “unionized” also means “non-ionized”
[Powers, 1995, ch. 7],
38
John Woods and Andrew Irvine
In its ordinary use in Greek, syllogismos can be translated as “computation” or
“reckoning.” In Aristotle’s logical writings, it is given a more technical meaning.
In its broad or generic conception a syllogism is an argument in which a conclusion
is derived of necessity from premisses, subject to further conditions to which we
shall recur. Syllogisms in the narrow sense are triples of categorical propositions
which are reducible to the first syllogistic figure as, for example, is the following:
All Greeks are human
All humans are mortal
Therefore, all Greeks are mortal.
Aristotle’s programme in the Prior Analytics has an ambitious objective. It is
to prove that all imperfect syllogistic forms reduce to the first syllogistic figure. 21
The very coherency of Aristotle’s mature programme requires that he have had a
sufficiently well-articulated conception of syllogisms in the broad sense to enable
the reductionist strategy to be judged.
There is, we say, a prior theory of syllogism in the broad sense. The theory is
presupposed by the Prior Analytics and is found in the Topics and On Sophistical
Refutations. 22 What is more, we find in the Topics a clear presentation of the
operation of argumental conversion, a subject to which we shall return in due
course. Also evident, as we have noted, is a well-managed distinction between the
relations of contradictoriness and contrariety, and early treatment of modus tollens
and modus ponens. The output of the Topics and On Sophistical Refutations serves
as input for the formalizing and metalogical devices of the Prior Analytics. It may
rightly be said that the inputs to the reductive devices of the Prior Analytics must
have structural features which enable the devices to engage them. In the Prior
Analytics these structural features can be thought of as logical forms. There is
no reason to suppose that the syllogisms of the early parts of the Organon lack
logical forms. What is true is that the theory of syllogism in the broad sense is
not a theory that manipulates those logical forms, at least overtly. We shall say,
therefore, that the theory of generic syllogisms is a pre-formal theory.
4 SYLLOGISMS IN THE GENERIC SENSE
Whatever else they are, syllogisms are valid arguments, or sequences of proposi¬
tions meeting certain further conditions: 23
21 Cf. Prior Analytics Al, 24 6 , 27; A23, 40 6 , 20.
22 In the Topics, syllogisms are discussed at 100°, 25-27. Cf. On Sophistical Refutations 164 6 ,
28 ff., and Rhetoric 1356 6 , 16-17; 1357“, 8 ff.; 1358“, 3 ff., among other places.
23 Even this is not quite without controversy. Aristotle reserves the term protasis for the
premisses of syllogisms. In fact, protasis is often translated as “premiss.” This leaves the question
of how to characterize conclusions. In as much as the conclusion of one syllogism might well be
the premiss of another, there is a reason to hold that a protasis is a proposition irrespective of
its role in any given syllogism.
Aristotle’s Early Logic
39
For a deduction [a syllogismos] rests on certain statements such that
they involve necessarily the assertion of something other than what has
been stated, through what has been stated (On Sophistical Refutations
165“, 1-3).
In several of the treaties of the Organon —for example in the Categories (see 2 a ,
35-2 6 , 7)—Aristotle attempts to bring forth an account of propositions. Inchoate
as it certainly is, and hardly consistent in all details, Aristotle’s treatment imposes
significant constraints on what is to count as a proposition. The core notion is that
in a proposition a single thing is predicated of a single thing. For example, in book
one of the Posterior Analytics a proposition is “one thing said of one thing” (72“,
9). Barnes suggests that this “one-one” principle, as we may call it, might have
been designed to rule out equivocal predications (see Metaphysics 4, 1006“, 32) or
multiple predications (see Topics I, 6; On Sophistical Refutations 169°, 8-9; 14-20;
181 a , 36-39 and On Interpretation , 18 a , 18-23). At On Interpretation 18°, 13-14,
Aristotle also writes that “a single affirmation or negation is one which signifies
one thing about one thing.” Barnes then directs us to a later passage (20\ 12-21)
where it is suggested that the “one-one” rule is designed to hold subjects and
predicates to the expression of metaphysical unities. On Sophistical Refutations
also has it that a proposition “predicates a single thing of a single thing” (169°,
7) and requires that “one must not affirm or deny several things of one thing nor
one thing of many, but [oidy] one thing of one thing” (181°, 38). Further, “since
a deduction starts from propositions and a refutation is a deduction, a refutation,
too, will start from propositions. If, then, a propoistion predicataes a single thing
of a singel thing, it is obvious that this fallacy [of Many Questions] too consists
in ignorance of what a refutation is; for in it what is not a proposition appears to
be one.” (On Sophistical Refutations 169°, 12-156; emphasis added). In a note to
Prior Analytics 24 a , 16-24 6 , 15, Smith also points out that Aristotle
developed a theory according to which every such sentence [i.e. propo¬
sition] either affirms or denies one thing of one thing, so that a single
assertion always contains a single subject and a single predicate. (In
On Interpretation , he always explains more complex sentences either as
having complex subjects or predicates or as really equivalent to groups
of sentences) [Smith, 1989, pp. 106-107].
It was not Aristotle’s intention to preclude plural propositions. What seems to
be meant is that declarative sentences cannot be propositions unless they are con¬
nective free , with the exception of something like predicate-negation. So, whereas
(a) All men are mortal
is a proposition,
(/3) All men are mortal or Socrates lives in Athens
and
40
John Woods and Andrew Irvine
( 7 ) All men are mortal and Madeleine lives in Vancouver
are not. On the other hand
(5) No men are non-animals
is a proposition.
There is a significant sense in which arguments such as that from (a) to (/?)
fail. Their failure does not consist in there being countermodels for them. They
fail in the theory of syllogisms on a non-deductive technicality. They either de¬
ploy or authorize the derivation of non-propositions, of statements that are not
propositions in Aristotle’s technical sense. Even so, the reason for the failure is
deductively salient. It permits, even if it does not invite, the conjecture that Aris¬
totle’s conception of validity is indistinguishable from our own. On this view, a
valid argument is any finite sequence of statements whose last member is necessi¬
tated or entailed by those that precede it. Further, since some statements are also
propositions in Aristotle’s sense, an argument is valid when its premisses entail
its conclusion, even when some of its statements are propositions and others not.
Where the validity rules fail, when they do, is in the theory of syllogisms. As we
may now say, a syllogism is a valid argument, all of whose statements are proposi¬
tions. Bearing in mind the translation of protasis as “proposition,” we propose to
call such arguments protaseic arguments. Thus the A-introduction rule fails in the
sense that no valid argument satisfying it can be a protaseic argument. Unlike va¬
lidity, the property of being a protaseic argument is not closed under the standard
deduction rules. (We are so using “valid” that an argument is valid just in case
its premisses necessitate its conclusion; and we are using “necessitate” to mean
what modern logicians mean by “implies” or “entails,” and these conventions will
remain in force unless otherwise indicated.)
Aristotle’s propositions in the technical sense will strike the modern reader as
something of a curiosity. What motivates so restricted a conception? On Inter¬
pretation bears directly on this question. There Aristotle advances the semantico-
grammatical thesis that all statements reduce to simple statements in ways that
preserve content. Simple statements are those that obey the one-one rule. Thus,
they are propositions in Aristotle’s technical sense. The thesis of On Interpretation
(17“, 13; 18“, 19 ff., 24) is that all statements reduce to propositions (assuming
reduction to be reflexive), and this we might call the thesis of propositional simplifi¬
cation. Thus “proposition” is a technical term for Aristotle, made so by the daring
thesis of propositional simplification. If the thesis is true, it is hugely important.
It isolates a sentential minimum adequate for the expression of all statements of
Greek.
Aristotle’s requirement that syllogisms be made up of propositions now seems
to be explicable. It greatly simplifies the task of specifying the class of syllogisms
and isolating their key properties. On this view, the propositional simplification
thesis achieves the same economies in the theory of syllogisms as it achieves in the
theory of grammar . 24
24 Cf. [Smith, 1989, p. 35]: “Therefore in studying categorical sentences [Aristotle] took
himself to be studying what can be said, without qualification. This last point is essential in
Aristotle’s Early Logic
41
A further indication of Aristotle’s motivation can be found in the
Topics, as we have said. Aristotle’s object, also stated at the beginning and
repeated at the conclusion of On Sophistical Refutations, is to discover a method
from which we will be able to syllogize about every issue proposed from endoxa,
i.e., reputable premisses, and, when compelled to defend a position, say nothing
to contradict ourselves (100 a , 20-22; 183 a , 37-183 6 , 6). A position to be defended
is called by Aristotle a problem, which he divides into four kinds, each correspond¬
ing to a different predicable. The four predictables are genus, accident, (unique)
property and definition. Every investigation of a problem involves determining
whether a predicable belongs to a subject as genus, as accident, as unique prop¬
erty or by definition. If it is characteristic of such predications that they involve
the attribution of one thing to one thing, it may be that Aristotle is embedding
this characteristic in his technical notion of proposition. Whatever the motivation,
the restriction to propositions is a fact about how Aristotle’s syllogisms are to be
constructed.
It is possible that Aristotle was influenced in his conception of an elementary
proposition by Plato’s contention ( Sophist 252C4 ff.) that a statement has min¬
imally a name ( onoma ) and a verb ( rhema). The function of a name is to refer
to something; but if we want to “get somewhere” (262D5) we must add to the
name a verb. Only then do we say ( legein) something. The result is a sentence
(logos) (262D5-6). Modern readers may see this as anticipation of Frege’s notion
of the unsaturatedness of predicates, since here too the utterance of a predicate
fails to “get somewhere” unless completed by a name or subject expression. In
On Interpretation Aristotle repeats the view that a logos is composed of an onoma
and a rhema , and no formula of whatever kind or degree of complexity is a sen¬
tence unless it contains a verb (On Interpretation, 17 a , 11-15). As stated, the
doctrine puts no obvious a priori limits on the complexity of names and verbs. At
On Interpretation 17 a , 39, it is implied that a subject term can be either general
(“man”) or singular (“Callias”)—but cf. I7 b , 3.
On the other hand, sentences whose predicates are singular terms or proper
names are not predications strictly speaking. They are ungrammatical. This
excludes would-be premisses such as “All wives of Socrates are Xanthippe,” even
though Xanthippe, in fact, is Socrates’ one and only wife. Also, names of accidents
may appear in predicate position but not in subject position. When an accident
name appears to occur in subject position it serves as the name not of the accident
but, rather, of the thing in which the accident inheres (Categories 5 b ). It is not
red that is coloured, but red things. So adjectives are admitted into the basic
onoma/rhema structure.
It is hard to see that these developments leave the “one thing” predicated of
“one thing” doctrine with any meaning except this: that propositions in what
we have been calling Aristotle’s technical sense are statements that conform to
understanding Aristotle’s theory of validity. In fact, this is a theory of validity for arguments
composed of categorical sentences, but since Aristotle thought that all propositions could be
analyzed as categoricals, he regarded the syllogistic as the theory of validity in general.”
42
John Woods and Andrew Irvine
the onomaf rhema structure of elementary sentences (hence, they have one name
of whatever degree of complexity and one verb of whatever degree of complex¬
ity) . “Whatever degree of complexity” of course is complexity consistent with the
one name/one verb structure. Thus “If Socrates is wise and Plato is wise, then
Socrates and Plato are wise” is disqualified, but not because it contains an ad¬
jective. Rather, it is because it contains a connective in virtue of which the one
name/one verb structure is violated.
It would be helpful at this point to revisit the claim that in the Topics and
On Sophistical Refutations syllogisms are inherently dialectical. Aristotle asserts
that the Topics is a genuinely original piece of work, forwarding conceptions and
insights that were entirely new. He repeats the point in On Sophistical Refutations:
Of the present inquiry, on the other hand, it was not the case that
part of the work had been thoroughly done before while part had not.
Nothing existed at all. ... If, then, it seems to you after inspection that,
such being the situation as it existed at the start, our investigation is
in a satisfactory condition compared with the other inquires that have
been developed by tradition, there must remain for all of you, our
students, the task of extending us your pardon for the shortcomings
of the inquiry, and for the discoveries thereof your warm thanks. (On
Sophistical Refutations, 183\ 34-184 6 , 8, emphases added).
Yet the Topics announces itself as concerned with a method “from which we
will be able to syllogize about every issue proposed from endoxa,” a method,
therefore, for the construction and presentation of dialectical arguments. Neither
dialectical nor refutation arguments are anything that originated with Aristotle.
Where, then, does the vaunted innovation of the Topics lie? Our view is that the
original contribution is the syllogism, developed in such a way as to elucidate the
deductive substructure of real-life arguments in their everyday uses as disputes
about received opinions, as arguments that refute an opponent’s claim, and so on.
If this is right, Aristotle lays claim to being the first systematic developer of applied
logic. That this is indeed right is suggested by the following considerations.
It is interesting to ask whether someone might be taught how to perform ar¬
gumentative tasks properly and efficiently, or offered guidance under which his
performance of them is improved. Aristotle’s answer is Yes. The Topics contains
a catalogue of propositions of possible use, in the sort of argument community that
Aristotle is addressing, in the derivation of target conclusions. Here the basic idea
is to find a set of acceptable propositions relevant to the issue under contention,
and the Topics attempts to give guidance on how to find such sets. There follows
a catalogue of rules and what might be called set-piece arguments (or schemas
of arguments) which take acceptable premisses to target conclusions. Bearing in
mind that Aristotle sometimes claims to be giving this guidance in such a way
that it can be followed by arguers who have no knowledge of the content or sub¬
ject matter of the disputed question, it is an audacious feature of the Topics that
it offers advice of a highly abstract nature, of a kind that might be described as
Aristotle’s Early Logic
43
“transcommunal,” that is, effective in an arbitrary community of arguers. For
example, let C be any target irrespective of its content; then the task is to find
premisses, Pi,...,P n , which, whatever their contents, are acceptable according to
acceptability criteria K\,...,K n , are relevant to C, and are such that C follows
from them. The employability of such rules for this task presupposes the possi¬
bility of recognizing the properties of premiss-acceptability, premiss-relevance and
premiss-consequence independently of premiss and conclusion content. If we were
intent on using such a strategy for the construction of a refutation, it must be
possible, first, to identify the thesis to be refuted. This is done operationally:
it is some proposition proclaimed by the one party, and which the other party
challenges. The refuter’s premisses in turn are acceptable if and only if they are
conceded by his opponent. They are relevant to the target conclusion, which is the
contradictory of his opponent’s thesis, if (loosely) they are about the same sort of
thing as the thesis in dispute. Further, they must satisfy the premiss-consequent
condition if a subset of those premisses consists of the premisses of a syllogism
for the negation of the disputed thesis. So there is a content factor here at work.
Arguers must know enough about the subject matter of their contention to know
whether a given would-be premiss is a proposition about that same subject mat¬
ter; but if their argument is being conducted in a topic-neutral way, they need not
understand that content.
In all cases, whether abstract or concrete, the overall approach of the Topics is
abductive. It seeks to answer the question, “What is the optimal set of premisses
from which to conclude a target conclusion?” The minimal answer is that a set of
optimal premisses is any set, 5, from which the target conclusion, C, is derivable.
The fuller answer not only cuts S down to a relevant and acceptable subset, but
it also cuts down the consequence relation in ways that we have yet to examine.
The Topics contains an abundance of (often confusing) instructions about how
to optimize the derivation of target conclusions. To this end suggestions abound
for premiss searches, and rules of derivation, as well as sample derivations, are
provided. Aristotle was scornful of the methods of the Sophists. He says, in effect,
that all that they offer the would-be arguer is set-piece arguments. Aristotle sees
nothing wrong with set-piece arguments, but on his view, they cannot constitute
an adequate methodology of successful argumentation. Catalogues of set-piece
arguments are deficient in two respects. They lack a systematic account of why
they are successful, if they are. Further, they lack systematic principles of extrap¬
olation to contexts and subjects of disputation for which the catalogue contains
no set-piece arguments as guides. It is in respect of these two deficiencies that
Aristotle’s claim to originality should be understood. In saying that the Topics
and On Sophistical Refutations constitute a wholly original innovation, that there
is something in these monographs that did not exist before, Aristotle invites us
to consider precisely those features that are absent from the sophist’s methodol¬
ogy and present in his own. Of course, there is a great deal in these works that
had existed before. There is the notion of dialectical argument, and of combative
or eristic variations of it, concerning which there is a huge preceding literature,
44
John Woods and Andrew Irvine
not least of which are the deep and detailed discussions of dialectical reasoning in
several of Plato’s dialogues ( Meno 86E-89C, Phaedo 95E7-107B, Republic 510D
511D, 527A6-B1, 533B-534D), and Zeno’s celebrated paradoxes, which Aristotle
took very seriously.
Then, too, it is a commonplace that when a conclusion is correctly derivable
from some premisses, there is a relation from premiss to conclusion in the absence
of which the derivation would not be correct. But it is certainly not true that
Aristotle was the first to recognize this commonplace. So, again, where does
the innovation lie? When we recall that Aristotle’s strategic rules include rules
for premiss searches and rules for the construction of derivations, it is clear that
Aristotle sees himself as specifying a type of argument whose conditions blend and
incorporate these two sets of rules. The type of argument in question Aristotle
calls syllogisms. The necessitation requirement is a condition on derivation. A
target conclusion is correctly derived from premisses only if it is necessitated (or
implied) by them. The requirement that a conclusion must not repeat a premiss is
a premiss-search (and premiss-eligibility) rule. The requirement that conclusions
be derived from and through (or because of) its premisses is another premiss-
eligibility rule, and so on.
On the face of it, this is not all that exciting. It can scarcely be imagined that
the definition of the syllogism would have struck any of Aristotle’s contemporaries
as a discovery: a useful tidying up of something commonly employed by disputants
perhaps, but surely not an original theoretical insight. The received wisdom in
our own time is that it was certainly not a discovery, or much of one anyhow, and
that the real innovations in Aristotle’s work in logic do not present themselves
until the Prior Analytics, what with its perfectibility result. But Aristotle was not
stupid, nor was he given to misplaced self-congratulation. When he says that there
is a wholly new theoretical twist to the Topics and On Sophistical Refutations it
would be a little short of insulting to ascribe this innovation to the definition of
Topics 100°, 25-27 and On Sophistical Refutations 165°, 1-3. Far more likely
is that Aristotle’s originality lies in the uses to which he is able to show that
syllogisms can fruitfully be put. That is, we may suppose that when Aristotle
wrote these treaties there were, in what might broadly be called the study of
argument, various open questions which no known account of argument was able
to handle satisfactorily. These included the following:
(1) When we refute someone, how can we be sure that our refutation is
correct, and how can we get the refutation to stick , i.e., at a minimum,
how can we guarantee our opponent’s acquiescence?
(2) When we argue against a position, how can we be sure not to have
begged the question against that position in our selection of premisses?
(3) Some people are of the view that argument is just word play and
clever self display, and that at bottom arguments do not get us any¬
where; they do not facilitate the realization of deep ends. Is this right
and, in particular, can argument ever lead to knowledge?
Aristotle’s Early Logic
45
(4) As any well-educated Greek knew, arguments abound in which the
conclusion is an utter violation of commonsense and deep scientific con¬
viction, but which seems with equal conviction to be correctly derived
from acceptable premisses. How are these paradoxical arguments to be
answered? How is the problem of pathological philosophy to be solved?
Aristotle’s innovation then consists in this: He is able to marshall, or so he
claims, the argumentative structures he has dubbed “syllogisms” in such a fashion
as to enable the satisfactory answering of each of these questions. Furthermore,
these answers are given in such a way as to reveal that their satisfactoriness de¬
pends indispensibly on features of embedded syllogistic structures. In this, as we
have said, Aristotle is the first applied logician. He is the first to show how answers
to these and other practical questions are rooted in what can only be called the
logical structure of deductive reasoning. For this to be true, it must also be true
that the definition of syllogisms is in some sense a surprisingly deep one.
We have said that few of Aristotle’s colleagues would have supposed the defi¬
nition to be all that deep, novel or surprising. What sharp contemporary Sophist
would have been bowled over by it? In fact, this is both right and wrong. It is
right in so far as the definition would strike the Sophist as intuitive and familiar.
It is wrong in so far as it turns out to be the case that structures defined by the
definition of syllogism have certain properties whose significance is not transparent
in the definition, and other properties whose existence is not transparent in the
definition. Here is a modern example, and a contentious one.
Someone might define the entailment relation in the “classically” semantic way:
$ entails $ just in case it is in no sense possible that $ and r -i4>* 1 are both the case.
On hearing it, people might say, “Of course,” or “Yes, that’s what it is all right.”
If the producer of the definition turned expectantly to his colleagues for praise
as an innovator, he would be disappointed. But suppose he went on to observe,
“Well, this being so, it follows impeccably that an impossible statement entails
every statement.” “Ah,” says a colleague, “your definition has hidden depths!”
Likewise, our task will not have been completed until it is shown how the syllo¬
gism facilitates Aristotle’s programme in applied logic. Without this connection,
it is open to a critic to complain that exposing the details of syllogistic struc¬
tures is conceptual complexity for its own sake, and that Aristotle has contrived
his account of syllogisms to no good end. In this context it is not our purpose
to emphasize Aristotle’s doctrine of paralogismoi. (But see [Woods and Hansen,
1997] and [Woods and Hansen, in progress].) Suffice it here to say that Aristotle
takes the paralogismoi of On Sophistical Refutations to be arguments that appear
to be refutations but, in fact, are not. A refutation is a syllogism meeting certain
specific conditions. Aristotle’s view is that in making, or accepting, a sophistical
refutation one commits the fallacy of mistaking it for such a syllogism. Since syllo¬
gisms are not inherently dialectical structures, and since they do inhere in the very
concept of a sophistical refutation, and of the fallacy that attaches to the making
or accepting of a sophistical refutation, the concept of fallacy is not exhausted by
merely dialectical factors.
46
John Woods and Andrew Irvine
If we have succeeded in showing that syllogisms are not inherently dialectical
structures, it is nevertheless left open for someone to claim that fallacies are inher¬
ently dialectical. We do not think that this is so, but if we are right, we face the
heavy weather of fallacies such as Begging the Question and Many Questions, each
of which, for the modern reader, is as dialectical as it gets. 25 Even so, except for
brief remarks in the section to follow we shall not, as we say, be much concerned
with this question.
However, before leaving this section, we shall say our piece about the similar¬
ities and dissimilarities between and among syllogisms, fallacies and sophistical
refutations. A syllogism is a valid argument meeting the additional constraints
we have already listed, together with others yet to be discussed. Thus the class
of syllogisms is a nonconservative restriction of the class of valid arguments. A
fallacy is an argument that appears to be a syllogism but which in fact is not. A
sophistical refutation is an argument that appears to be a refutation but which
in fact is not. When an argument merely appears to be a refutation it owes this
appearance to the fact that it embeds something that appears to be a syllogism
but is not, or to the fact that it embeds a syllogism whose conclusion appears to
be, but is not, the contradictory of the original thesis whose refutation is sought.
Thus an argument is a sophistical refutation when it appears to be a refutation
but embeds either a fallacy or a non-fallacy with the wrong conclusion.
Real-life arguments involve more than the production of syllogisms. With refu¬
tations as an example, there are also constraints on how premisses are selected.
In this case, the refuter must draw his premisses from concessions given by an
opponent in answer to the refuter’s Yes-No questions. Or, as another example,
demonstrations consist of syllogisms whose premisses must be drawn from the de-
scendent class of a science’s first principles under syllogistic consequence. In both
cases there is more to the real-life argument than a mere sequence of premisses
and conclusion. In each case there are additional conditions on premiss-eligibility.
These are not themselves syllogistic conditions. This enables us to see that an ar¬
gument might be a perfectly good syllogism and yet be a perfectly bad refutation
or demonstration (or instruction argument or examination argument). If we think
of the syllogism embedded in a real-life argument as its strictly logical component,
then it is clear that most real-life arguments also satisfy non-logical constraints.
It is also clear that in some cases, but not all, these non-logical constraints include
conditions that can be called dialectical in ways that we have been considering.
25 Thus we have [Hamblin, 1970, pp. 73-74]: “The Fallacies of Begging the Ques¬
tion and Many Questions depend in conception, more than any other kinds, on the con¬
text of contentious argument ... The Fallacy of Many Questions can occur only when
there is actually a questioner who asks two or more questions disguised as one.” See also
[Hintikka, 1987, p. 225]: “[0]ne thing is clear of the so-called fallacy of many questions. It
cannot by any wildest stretch of the imagination be construed as a mistake in inference. It will
thus bring home to the most hardened skeptic the impossibility of seriously construing Aris¬
totelian fallacies in the twentieth century sense, i.e., as tempting but invalid inferences.” But
c/.: “It is not clear in Aristotle’s writings that the so-called fallacy of many questions is thought
of by him just as a violation of presuppositions of questions” [Hintikka, 1987, p. 224] (emphasis
added).
Aristotle’s Early Logic
47
5 WHY THE FALLACIES ARE IMPORTANT
It may strike some readers as odd that Aristotle develops his generic account of the
syllogism to stabilize the distinction between good arguments and good-looking
arguments. In the absence of such a distinction, a general theory of argument
would certainly be significantly disabled. Aristotle’s optimism may incline us
to think that the theory of syllogisms now makes this a usable and principled
distinction, and that the theory of argument can now proceed apace. Yet clearly
this would be to misjudge Aristotle’s own view of the matter, as is evidenced by
his very concept of fallacy.
As we have said, in its broadest sense a fallacy is something that appears to be
an argument of a certain type but which, in fact, is not an argument of that type.
In its use in On Sophistical Refutations, a fallacy is an argument that appears
to be a syllogism but is not, in fact, a syllogism. We see, then, that the concept
of syllogism is bedevilled by the same uncertainty that affected the more general
concept of argument. Aristotle thinks that a good argument is one that is, or
subsumes, or is in some other way intimately related to, a syllogism. But just
as it is not always possible to distinguish a good argument from a good-looking
argument, we also have it that it is not always possible to distinguish between a
syllogism and something that only looks like a syllogism. Syllogisms were to be
the means of removing the indeterminacy between good and merely good-looking
arguments. Yet syllogisms are afflicted by this self-same indeterminacy. How,
then, can syllogisms perform their restorative function in the general theory of
argument?
Aristotle will overcome this problem in what rightly can be said to be the great¬
est technical achievement of the Prior Analytics, namely, the (almost sound) proof
of his perfectibility thesis. Aristotle distinguishes between perfect and imperfect
syllogisms. It is an oddly expressed distinction in as much as it is not the case that
an imperfect syllogism is any less a syllogism than a perfect one. What Aristotle
intends to capture with this distinction is the contrast between arguments that
are obviously syllogisms and arguments that, while they are syllogisms, are not
obviously so. According to the perfectibility thesis, there is, for any imperfect
syllogism, a perfect proof that the argument in question is a syllogism. A perfect
proof is one, all of whose rules are obviously good rules. There are two types of
perfect rule. One, which Aristotle calls common, are rules such as conversion and
reductio ad absurdum. The other type of perfect rule, for which we propose the
name syllogistic rule, is the conditionalization of any perfect syllogism. Finally,
in a proof of the perfectibility of an imperfect syllogism, the original argument’s
premisses serve as hypotheses of a conditional proof. To these hypotheses, perfect
rules are applied to generate conclusions which may themselves serve as hypothe¬
ses to which perfect rules may also be applied. The conditional proof terminates
with the derivation of the original conclusion of the imperfect syllogism. Thus
a perfectibility proof is a conditional proof of the original argument’s conclusion
from the original argument’s premisses by repeated application of perfect rules.
48
John Woods and Andrew Irvine
According to the perfectibility thesis, the conclusion of any imperfect syllogism is
in the descendent class of the argument’s premisses under the perfect rules. In this
way, what Aristotle claims to have demonstrated is that for any syllogism that is
not obviously a syllogism, there exists a perfectly perspicacious way of making it
obvious that the argument in question is a syllogism.
The perfectibility thesis is discussed in greater detail in the Prior Analytics.
We mention it here to make the point that since it was not something that Aris¬
totle could draw on in his earlier writings, the issue of fallacies remains a serious
difficulty for the earlier logic. This makes it all the more curious that Aristotle’s
treatment of the fallacies is, for the most part, rather thin and fragmentary. As
long as it remained the case that fallacies could not be recognized in a principled
way, then the invention of logic itself would exacerbate the very problem it was de¬
signed to solve. For as long as we lack a principled grasp of the distinction between
syllogism and fallacy, syllogisms can play only an uncertain role in distinguishing
between good and bad arguments. When we return to a discussion of the fallacies
in section 12, it will be advisable to keep this point in mind. It helps in attaining
an understanding of Aristotle’s problem-solving methodology.
Given that Aristotle’s problem is to distinguish between syllogisms and fallacies,
it is clear that Aristotle has two general strategies to consider. One is to produce
what in fact he never got around to producing, namely, a full account of each of
the fallacies in the original taxonomy of thirteen, and of those other fallacies (such
as ad hominem ) mentioned elsewhere. 26 But a second possibility is that Aristotle
would hit upon a way of making syllogisms effectively recognizable, which would
not require an account of any fallacy, whether fragmentary or full. As we have
remarked, some writers ( e.g ., [Boger, 1998a]; cf. chapter 3 of this volume) are
drawn to the view that the logic of the Prior Analytics was already available to
Aristotle when he was writing about the fallacies in On Sophistical Refutations.
We ourselves tend to demur from this opinion largely for reasons set forth in
[Hitchcock, 2000a]. But it is grist for the mill of this controversy that in what we
take to be his earlier writings on logic, Aristotle expressly avails himself of neither
strategy. We could go so far as to say that in On Sophistical Refutations the
fallacies elude Aristotle’s theoretical grasp and, indeed, his theoretical interest. If
we were to take this latter possibility seriously, we would be left with the necessity
of trying to explain how it came to pass that having exposed a gaping wound in the
theory of syllogisms Aristotle had no interest in following this up in a theoretically
determined way. One possibility is that he was stymied, and did not yet know how
to proceed with the requisite theoretical articulation. The other is that he already
had a conception of how he would proceed in the Prior Analytics. If Boger is right,
he had this conception of how he would proceed in the Analytics because that way
of proceeding was already an accomplished fact during the writing of Topics and
On Sophistical Refutations.
26 Care needs be taken in attributing to Aristotle the view that ad hominem arguments are
fallacies. In one sense of “proof’ they are proofs of no kind; but in another sense of “proof,”
they are proofs of that kind. See, below, section 11 and [Woods, 2003, ch. l].
Aristotle’s Early Logic
49
Whichever explanation is favoured, it is worth noting that from the point of
view of the syllogism, Aristotle’s examples of sophistical refutations very often do
not even appear to contain syllogisms. In some places, for example, quantifiers are
conspicuous by their absence (On Sophistical Refutations 165 6 , 34-35 and 166°,
10-12). In others the number of premisses is wrong ( e.g ., 166 6 , 37; 168 a , 12-
16; 168 6 , 11; 180“, 33-34). In still others, premisses and conclusions are not in
strict propositional form (e.g., 165 6 , 38-166“, 2; 166°, 9-10; 167°, 7-9, 29-30;
167 6 , 13-17; 177“, 36-38; 177\ 37-178“, 2, 11-16; 178 6 , 24-27; 179“, 33; 180“,
34-35; ISO 6 , 9-10, 11-12, 21-23 and 23-26). 27 What these deviations suggest to
us is that Aristotle’s interest in these examples is a good deal more everyday than
theoretical. He is more interested in getting across the main ideas of his taxonomy
of fallacies that ruin refutations than showing in strict detail that they instantiate
non-syllogisms that really do appear to be syllogisms.
6 A LOGIC OF GENERIC SYLLOGISMS
In saying that Aristotle’s notion of syllogisms in the broad sense is a contending
contribution to logical theory, it is necessary to have in mind some fixed star with
which to box our compass. We shall need to have in mind a conception of what
a core logic is. It is widely assumed by present-day theorists that the core of
logic is the study of deducibility relations and that these relations display three
jointly sufficient structural properties that capture the essentials of the deductive
transmission of information. These three properties are reflexivity, transitivity
(also called cut), 28 and monotonicity (also called dilution). 29
By reflexivity any statement is derivable from itself, or (to have an expositionally
handy converse) yields itself. By transitivity any statement yielding a statement
which itself yields another also yields that other. By monotonicity any statement
derivable from a statement is also derivable when that statement is supplemented
by any others in any finite number. On the view we are examining, a core logic
is a theory of deducibility in which the deducibility relation satisfies these three
structural conditions. We call such a logic a Gentzen logic.
As will become apparent, Aristotle’s conception of the syllogism fails the Gentzen
conditions hands down. This indicates that Aristotle would be sympathetic to a
distinction which the expression “deducible from” all but obliterates. This is the
distinction between implication and inference. Concerning inference, it appears to
be Aristotle’s position that while inference may obey (perhaps a restricted form of)
transitivity, it certainly does not obey either reflexivity or monotonicity. In con-
2 'These and other syllogistic deviations are well canvassed in [Hitchcock, 2000a].
28 This is not quite accurate. Transitivity requires that the subordinate argument have only
one premiss, which is identical to the conclusion of the superordinate argument. Cut permits
multi-premissed subordinate arguments, where one of the premisses is identical to the conclusion
of the superordinate argument.
29 These properties are proclaimed in the three structural rules of the same name of Gent.zen’s
sequent calculus. See [Gentzen, 1935]; [Szabo, 1969]. See also [Scott, 1971, p. 133].
50
John Woods and Andrew Irvine
trast, Aristotle’s idea of implication is given by his notion of necessitation, which
is an unanalyzed primitive in his writings [Lear, 1980, pp. 2, 8]. We will suggest
in due course that necessitation should be understood as fulfilling the Gentzen
conditions, hence that Aristotle has, implicitly, a core logic for the implication
relation.
Part of what may be truly original about Aristotle’s thinking is its apparent
openness to a twofold fact: first, that inference is not (the converse of) implication
but, second, that inference can be modelled in a restriction of the core logic of
implication. So conceived, the inference relation is the converse of the implication
relation under certain rather powerful constraints. What is more, a valid deduction
in a Gentzen logic, when subjected to those same constraints, yields a structure of
a sort that Aristotle called syllogisms. This suggests that what Aristotle wanted
to do with the concept of syllogism was to “inferentialize” the validity rules of a
given core logic. That is, he wanted to make rules such as Gentzen’s deducibility
rules more like rules of inference.
We trust that we will not have to apologize for anachronisms so blatant as to be
self-announcing. Charity, if nothing else, provides that brazenness alone cancels
any idea of express attribution to Aristotle. But Aristotle does have an implication
relation rolling around in his theory of syllogisms, and we should want to know
what it is. We are saying that if it is the implication relation of a Gentzen logic,
then we get the result just noted. Of course, it may strike us as obvious that
getting this result is nowhere close to showing that Aristotle’s implication is the
converse of Gentzen-deducibility and that Aristotle’s validity is Gentzen-validity.
We should think again. Gentzen-deducibility and Gentzen-validity are structures
of a core logic; i.e., they satisfy the three structural rules of reflexivity, transitivity
and monotonicity. Of course, in Gentzen’s own calculi the structural rules are
supplemented by what Gentzen called “operational rules,” and these are rules
which, under certain assumptions, characterize the logical constants. What we
are saying here has nothing to do with operational rules. We are not saying that
Aristotle’s validity is the validity of the full sequent calculus. We are saying only
that a case can be made for supposing that Aristotle’s validity is validity according
to the core properties, that is, validity as characterized by these three structural
rules. Here is the case. Whatever its details, the property of being a syllogism,
or “syllogisity,” is some kind of validity, minus the properties of reflexivity and
monotonicity.
There is a crucial difference between syllogisity and Aristotle’s validity, what¬
ever it is in detail. Syllogisms are irreflexive and nonmonotonic. Let V be a
property of arguments that results from the syllogisity property by reimposing the
conditions of reflexivity and monotonicity. Then V is (core) Gentzen-validity if
V is also transitive. But it is plausible to suppose that nothing qualifies as valid¬
ity unless it obeys transitivity; so if V is transitive, then since it is also reflexive
and monotonic it is core Gentzen-validity. And since Aristotle’s definition of syllo¬
gisity implies that syllogisms are valid arguments, Aristotle’s validity, whatever its
details, is on this plausible assumption transitive. So there is Aristotle’s validity
Aristotle’s Early Logic
51
property—call it V a —which, being a validity property, is transitive on our present
supposition. And there is Aristotle’s syllogisity property to which, when reflexivity
and monotonicity are restored, gives V. But V just is V a . So Aristotle’s validity
is core Gentzen-validity. Thus, syllogisms are Gentzen-valid arguments for which
the conditions of reflexivity and monotonicity are stipulated to fail; and, equiva¬
lently, syllogistic implication is Gentzen-implication failing those some conditions.
If this is right, it is important. For in one good sense of the word, syllogisms have
an underlying core logic. Let us look to this possibility in greater detail. Let us
attend to syllogisms.
Aristotle says that “a refutation is a syllogismos ” {On Sophistical Refutations
1, 165“, 3). 30 This is a view in which he clearly persists, for it is repeated in the
Prior Analytics: “Both the demonstrator and the dialectician argue syllogistically
after assuming that something does or does not belong to something” (Al, 24“,
26-27), and
it is altogether absurd to discuss refutation without first discussing
syllogismos ; for a refutation is a syllogismos , so that one ought to
discuss syllogismos before describing false refutation; for a refutation
of that kind is a merely apparent syllogismos of the contradictory of a
thesis {On Sophistical Refutations 10, 171“, 1-5).
Let us, then, “first discuss syllogisms”:
A syllogismos rests on certain statements [i.e., propositions] such that
they involve necessarily the assertion of something other than what has
been stated, through what has been stated {On Sophistical Refutations
1, 165“, 1-3).
This is very much Aristotle’s long-held and settled view. The same conditions are
laid down at Topics 1, 100“, 25-27, 31 and repeated in the Prior Analytics Al, 24 6 ,
19-20:
A syllogismos is a discourse in which, certain things being stated, some¬
thing other than what is satated follows of necessity from their being
so.
Further, says Aristotle,
I mean by the last phrase that it follows because of them, and by this,
that no further term is required from without in order to make the
consequence necessary {Prior Analytics, 24 b , 20-22).
30 Unless otherwise noted all translations are from [Barnes, 1984], An exception is ai/Wo'yipos
translated by Barnes as deduction, but for which we have used the transliteration syllogismos.
31 “A syllogismos is an argument in which certain things being laid down, something other than
these necessarily comes about through them.”
52
John Woods and Andrew Irvine
Syllogisms, here, are what Aristotle calls “direct.” They contrast with “hypo¬
thetical syllogisms” , 32 which we shall not be much concerned with in these pages.
It suffices to remark en passant upon an interesting feature of the distinction
between direct and non-direct syllogisms. Hypothetical syllogisms, in contradis¬
tinction to those of the direct variety, are arguments construable as indirect proofs
in modern systems of natural deduction. In typical cases, they are per impossibile
arguments, that is, arguments such as the following:
Given:
(1) All A are B
Premiss
(2) Some A are not C
Premiss
To prove: (K) Some B are not C.
(3) All B are C
Hypothesis, contradicting K
(4) All A are C
From (1), (3)
(5) (4) contradicts (2)
Thus: ( K ) Some B are not C.
A key difference between direct and indirect proofs is reflected in the different
roles played by propositions introduced as premisses and propoistions introduced
as hypotheses. Premisses are permanent in all arguments in which they occur.
Hypotheses have a fugitive role. They are introduced, they perform their in¬
tended functions, then they are cancelled. A simple way, therefore, of marking the
distinction between direct and hypothetical syllogisms is to notice that in direct
syllogisms all propositions other than the conclusion must be premisses, whereas
in hypothetical syllogisms at least one such line must be a non-premiss, that is, a
hypothesis.
Aristotle concedes that the perfectibility proof of the Prior Analytics applies
only to direct syllogisms ( Prior Analytics, 41°, 37-41 6 , 1). If so, hypothetical
syllogisms are truly a breed apart. In a way, this is an ironic twist. As we saw,
Aristotle’s project is to perfect all syllogistical reasoning. Perfection is achieved by
reduction to the first figure. Aristotle recognizes that in some cases, the reductions
can be indirect, by way of arguments per impossibile ; and he says further that all
reductions whatever are achievable in this way ( Prior Analytics, 62 6 , 29-31; 41°,
23-24). But per impossibile arguments are hypothetical syllogisms. Thus some
of the syllogisms used by Aristotle to show that all syllogisms reduce to the first
figure are themselves syllogisms which do not reduce to the first figure. 33
32 [Lear, 1980, ch. 3; pp. 34-35]. The last chapter of the Topics is a struggle to get clear about
hypothetical syllogisms, and the need to do so is evident in the discussion of refutations in On
Sophistical Refutations. The task is taken up again in the Prior Analytics at 40 6 , 22-26; 41“,
23-26, 32-37; A44, 50“, 16-28.
33 Ironic though the twist may be, neither it nor its irony is lost on Aristotle in the Prior
Analytics. At A2, 25“, 14-17, there occurs a proof of e-conversion concerning which “all scholars
Aristotle’s Early Logic
53
In one respect the analysis of direct syllogisms is a matter of lively controversy.
Some writers hold that they are irreducibly conditional in form, hence that they are
a kind of statement . 34 Others are of the view that they are argumental structures,
hence sequences of statements . 35 Others, still, favour the ecumenical suggestion
that they can be taken either way and that the two approaches are interderivable
without significant loss . 36 Not wanting to re-open this debate, let us simply declare
for the second alternative. Aristotle’s syllogisms are structures of a sort that a
modern reader would identify as derivations in a system of natural deduction . 37,38
A syllogismos, then, “rests on certain statements such that they involve neces¬
sarily the assertion of something other than what has been stated, through what
has been stated.” As we see, syllogisms are thus valid sequences of propositions,
distinguished as to premiss and conclusion, which satisfy the following two condi¬
tions:
agree that Aristotle’s argument is ecthetic [i.e., not narrowly syllogistic]” [Mignucci, 1991, p. 11].
An argument is ecthetic if Darapti is proved ecthetically (see A6, 28“, 22-26); and at A8, 30 a , 6-
14, ecthetic arguments are advanced for Baroco NNN and Bocardo NNN, each a modal syllogism.
Moreover, the proof of Darapti requires modus ponens [Mignucci, 1991, p. 23] and the proof of
Baroco NNN requires modus tollens. Aristotle expressly recognizes that neither modus ponens
nor modus tollens is reducible to syllogisms in the narrow sense ( Prior Analytics A23, 41“, 23 ff.;
A44, 50“, 16 ff.). On the other hand, Aristotle also claims that any conclusion sanctioned by a
per impossibile syllogism can also be derived by a direct syllogism employing the same premisses.
34 See, for example, [Lukasiewicz, 1957, pp. 20-34] and [Patzig, 1968, pp. 3-4].
35 For example, [Smiley, 1973]; [Corcoran, 1972]; [Lear, 1980, pp. 8-9]; and [Frede, 1987, pp.
100-116],
36 Cf. [Thom, 1981, p. 23]: “Aristotle’s syllogistic can ... be presented, either as a system
of deductions [arguments] (a natural deduction system) or as a system of implicative theses
[conditionals] (an axiomatic system). [Smiley, 1973] has carried out the former task admirably
well; we shall attempt the latter. But, for those who remain unconvinced that the syllogism can
be treated as an implication, we shall provide a way of re-interpreting our system as a natural
deduction system.”
37 Here is Corcoran on the point: “My opinion is this: if the Lukasiewicz view [that
Aristotle’s logic is an axiom system] is correct then Aristotle cannot be regarded as
the founder of logic. Aristotle would merit the title no more than Euclid, Peano or
Zermelo insofar as these men are regarded as founders, respectively, of axiomatic geometry,
axiomatic arithmetic and axiomatic set theory. (Aristotle would merely have been the founder
of “the axiomatic theory of universals’)” [Corcoran, 1974b, p. 98].
We note in this connection that Gentzen’s structural rules are not by any means exclusive to
the Gentzen calculi. They hold in Frege’s system and in virtually every other logic published
subsequently. Why do we invoke the name of Gentzen? Why is the core theory of validity not a
Frege-logic or a Whitehead-Russell-logic? Our answer is that Gentzen was the first (along with
Jaskowski, independently) to break with the axiomatic tradition in modern logic and to show
that natural deduction systems have all the power of axiomatic set-ups. Because we hold, with
Corcoran, that Aristotle conceived of logic in natural deduction terms, it is seemly to use the
honorific “Gentzen” in reconstructing Aristotle’s conception of validity.
38 Terminological Note: We are using the expressions “deducible from,” “consequence of’ and
“follows from” without due regard for what logicians have come to admire in a distinction between
logical syntax and semantics. Even in the absence of a theoretically weighty divide between
syntax and semantics, there is an intuitive distinction between deriving a conclusion from certain
premisses and that conclusion following from them. We return to this point at the conclusion of
the present section. For now we shall only say that any looseness in our usage will be tightened
by context.
54
John Woods and Andrew Irvine
Min: They are minimal, that is, they contain premisses needed for
their validity and none other.
Non-Circ: They are elementarily non-circular, that is, their conclu¬
sions repeat no premiss.
Condition Non-Circ comes directly from this characterization of syllogisms. The
conclusion of a syllogism is something other than what has been stated, that is, its
premisses. There are two ways in which an argument might violate Non-Circ. Its
conclusion might repeat a premiss exactly as formulated, “word for word,” or its
conclusion might be a form of words syntactically different from a preceding line
but synonymous with it. Assuming the reflexivity of synonymity, the two cases sum
to one in the requirement that the conclusion of a syllogism not be synonymous
with any premiss. It is sometimes supposed that circularity is a species of question¬
begging. We believe this is not Aristotle’s own view. But whether it is or not,
it is not Aristotle’s intention to impose on syllogisms the requirement that they
not beg questions. Whether an argument begs a question or not arises only in the
context of further conditions which a syllogism might fulfill, as when it is used
as a refutation of an opponent’s thesis. When it is so used, it is held to a non-
question-beggingness constraint, but it is a constraint not on syllogisms as such
but on refutations, i.e., on syllogisms in their use as refutations. We employ in
Non-Circ the qualification “elementarily” to mark this point. An argument that
fails to be a syllogism because of its violation of Non-Circ is one in which the
conclusion is synonymous with some premiss, and hence repeats it. It is clear that
Non-Circ denies syllogisms the property of reflexivity, as witness the argument
A II- A. Even if it is allowed that A necessitates A, it could not be true that
A Ih A is a syllogism, since the conclusion A repeats the premiss A. So, syllogistic
implication is not reflexive.
Premisses will “involve necessarily” propositions other than what has been
stated by the premisses. “Involve necessarily” has the sense of “ following of ne¬
cessity” (Prior Analytics Al, 24 b , 20-22). From this it can be seen that Aristotle
requires that syllogisms be protaseic arguments, that is, that their premisses en¬
tail their conclusions. This is given in the basic condition that a syllogism is a
protaseic argument, that is, a valid argument all of whose statements are propo¬
sitions. Min makes the additional point that if a valid argument is a syllogism it
cannot contain superfluous premisses. For an argument to be a syllogism it is not
enough that its conclusion results of necessity from other propositions but, rather,
that the conclusion results of necessity because of them. 39 It is open to question
whether premiss-minimality captures all there is of the “because-of-them” require¬
ment. We shall not pursue the matter here, but will return to it later. We see that
the property of being a syllogism (again, “syllogisity” for short) is not a monotonic
property. It is consistent to suppose that necessitation is monotonic, but syllo¬
gisity is a restriction on necessitation (or validity). If syllogisity were monotonic
39 Cf. Posterior Analytics 7\ b , 22: Premisses must be “causes of the conclusion” [Ross, 1949]
or “explanatory of the conclusion” [Barnes, 1984],
Aristotle’s Early Logic
55
then, if A, B lb C were a syllogism, so too would be A,B,D\,...,D n lb C, for any
Di. But A, B, D\,D n lb C offends against Min. It is not a syllogism even if it
is a valid argument. Since validity is reflexive, every statement is validly deducible
from just one premiss, namely, itself. Non-Circ denies this property to syllogisms.
Various commentators have read off further conditions from our present two.
Mindful of syntactic niceties, it has been proposed that the plural form “certain
things being supposed” precludes single-premissed syllogisms , 40 and that the sin¬
gular form of “thing which results” rules out multiple conclusions. Corcoran’s
opinion is that Aristotle did not require of syllogisms as such that they have just
two premisses. That he did not impose this restriction
is suggested by the form of his definition of syllogism ([Prior Analytics]
24 b , 19-21), by his statement that every demonstration is a syllogism
(25 b , 27-31; cf. 71 b , 17; 72 b , 28; 85 6 , 23), by the context of chapter 23 of
Prior Analytics I and by several other circumstances .... Unmistakable
evidence that Aristotle applied the term in cases of more than two
premises is found in Prior Analytics I, 23 (especially 41°, 17) and in
Prior Analytics II, 17, 18 and 19 (esp. 65 6 , 17; 66 “, 18 and 66 6 , 2)
[Corcoran, 1974b, p. 90].
Still, it is clear that Aristotle often does reserve the term “syllogism” for two-
premiss arguments. We follow Corcoran in supposing that such a restriction is
explicable by the fact that Aristotle thought if all two-premiss syllogisms are de¬
ducible in the logic of the Prior Analytics, then all direct syllogisms whatever are
also deducible.
On the other hand, evidence from the Topics plainly indicates Aristotle’s will¬
ingness to countenance syllogisms of just one premiss containing two terms not
occurring in the conclusion [Allen, 1995, p. l]. There is, in any case, little doubt
that the settled opinion is that syllogisms require at least two premisses. We shall
tentatively record the consensus in a further condition, viz.,
Prem+: They are multi-premissed. 41
40 See John Maynard Smith, “Notes to Book A,” [Smith, 1989, p. 110]. See also [Frede, 1987,
p. 114]: “The Greek commentators all agree that the plural of ‘certain things being laid down’
has to be taken seriously as referring to a plurality of premises ... and everybody in antiquity
(except for Anipaster, cf. Sextus Empiricus P.H. II, 167) agreed that arguments have to have at
least two premises.”
However, as Barnes points out, there is textual evidence that Aristotle plumped for premisses
greater than two [Barnes, 1975, p. 68]. See Prior Analytics A14, 34“, 17-18; 23, 40 & , 35; and
Posterior Analytics I 3, 73°, 7-11. But see Prior Analytics 42“, 30-34: “So it is clear that every
demonstration and every deduction will proceed through three terms only. This being evident,
it is clear that a conclusion follows from two propositions and not from more than two....”
41 Against Prem+ , Robin Smith writes: “Aristotle thinks this is worth arguing for; but if,
as the ancient commentators thought, it is simply part of the definition—implicit in the plural
‘certain things being supposed’—then the point is trivial and the argument redundant” [Smith,
1995, p. 30], But this overlooks the fact that it is an open question for Aristotle whether indeed
definitions can be argued for. This he discusses in the Topics and Prior Analytics (and comes
up with contradictory answers).
56
John Woods and Andrew Irvine
Also advanced is the view, which we share, that a premiss for a syllogism is
the result of a distillation from all those contexts [of conversational use]
of a fundamental core meaning, excluding any epistemic [and semantic]
properties ... [Smith, 1995, p. 108]. 42
Thus the premisses of a syllogism are the propositional contents of cognate
speech acts ( e.g ., asserting that P, asking whether P, etc.) independently of how
they are spoken, independently of whether they are true, and independently of
whether they are known in a certain way, or at all. 43 Syllogistic premisses are, in
Aristotle’s technical sense, propositions. In their turn, syllogisms are sequences of
these, themselves independent of the pragmatic, semantic and epistemic conditions
of their production in day-to-day social congress. If this is right, let us again note
that syllogisms are not inherently dialectical structures; they are inherently logical
structures. 44 They are protaseic arguments satisfying the restriction-conditions
Min, Non-Circ and Prem+.
We are now launched on the task of discerning an account of syllogisms in
the broad sense in Aristotle’s dialectical treatises. The core conception is set by
conditions Min, Non-Circ and Prem+, with Prem-h admitted on sufferance. They
seem to follow fairly immediately from a characterization of syllogisms sufficiently
recurrent in the Organon, and beyond, to warrant the status of a definition. If these
conditions are taken as our theoretical basis, then the theory of syllogisms will be
roughly the least class satisfying Min, Non-Circ and Prem+, supplemented here
and there by various other textual implications. We must proceed to the further
specification of the theory, and we will do so with an eye on the collateral obligation
to judge it as a contending theory, i.e. , a theory that merits our consideration in
its own right, apart from its antiquarian importance. But there are other things
to do first.
42 Cf. [Lear, 1980, p. 51]: “A direct syllogism may be described in an epistemic vacuum. One
may or may not know the premisses and one may or may not use a knowledge of the premisses
to gain knowledge of the conclusion.” See also [Frede, 1987, p. 110]: “But later Peripatetic
authors, and even Aristotle in the Analytics, no longer thought of the definition [of ‘syllogism’]
as dependent on dialectical context.” Cf. Prior Analytics A32, 47“, 33-35. It is well to note that
the semantic independence of syllogisms is independence from the truth of their premisses; it is
not independence from the entailment of their conclusions.
43 If this sounds too Fregean a view for the likes of Aristotle, it suffices to characterize premisses
as declarative sentences considered in isolation of contexts of their use.
44 The point is sharpened by contrasting Aristotle’s definition of “syllogism” with, say,
Boethius’. Boethius says that a syllogism is an expression ( oratio ) in which when some things
have been laid down ( positis ) and agreed to ( concessts ), some different things follow necessarily
by virtue of the things which were agreed to (De Differentus Topicis , ed. Patrologia Latina, vol.
64, coll. 1173-1216). “[MJediaeval commentators explain this divergence from Aristotle by saying
that Boethius defines the dialectical syllogism, and Aristotle the syllogism as such. This cannot
be historically correct, however, since Boethius makes the same ‘addition’ in his De Categoricis
Syllogisms, and there it is certainly the syllogism as such that he defines. Thus it seems that
Boethius demands that the premises of a syllogism are accepted as true” [Green-Pederson, 1984,
pp. 44-45], emphasis added.
Aristotle’s Early Logic
57
For example, we want here to revisit briefly the claim that syllogisms are not
inherently dialectical structures. If this is so, then Aristotle’s fallacies are not
inherently dialectical either. In its most general sense, Aristotle thinks of a fallacy
as something which appears to be a good argument of a certain kind, but which is
not in fact a good argument of that kind. So understood, there are several ways
in which an argument could be a fallacy. Its premisses might not necessitate its
conclusion, though they appear to. It might contain an inapparent redundancy
in its premiss set. Its conclusion may be identical to one of its premisses in
camouflage. It may be an argument that appears, but fails, to be a demonstration,
i.e., a syllogism from first principles; and so on.
If a fallacy is something which appears to be a good argument of a certain
kind, but is not, there might be (and are) fallacies which merely appear to be
good dialectical arguments. Aristotle understands dialectical arguments to be
arguments from dialectical premisses, and he understands dialectical premisses to
express opinions either widely held, or supported by experts, or endorsed by “the
wise” (i.e., endoxa). One way, then, for an argument to be a fallacy is by being a
syllogism from premisses which appear to be endoxa but are not. Were it the case
that fallacies as such are dialectical, it would have to be true that all arguments
that merely appear to be good do so because they contain premisses that merely
appear to be reputable. But this was never Aristotle’s own view (c/. Topics 101“
ff.). By this same reasoning we should also resist the idea that the thirteen types
of sophistical refutation listed in the treatise of the same name are inherently
dialectical. In strictness, the thirteen are types of fallacies, hence types of ways in
which arguments can be sophistical refutations; but an argument is a sophistical
refutation when it merely appears to be a refutation, and this clearly can happen
even when its premisses appear to be, and are, expressions of reputable opinion.
There is value in having made this aside, for it highlights a touchy ambiguity
in the concept of dialectical argument. The idea of dialectic is in fact multi¬
ply ambiguous in Aristotle’s thought, and in Greek philosophy generally. What
matters here is an ambiguity that straddles Aristotle’s thinking and that of his
present-day successors. For writers such as Hintikka and Hamblin, an argument
is a multi-agent interchange of speech acts 45 over which, however inchoately, the
idea of a challenge is definable. In some sense, parties to a dialectical exchange
are each other’s opponents. Typical is the question-answer dialogue in which one
party seeks to refute a claim of another party. Here the notion of challenge is
overtly applicable, but we may also find it in muted form in the interrogative
exchange between teacher and pupil, in which there are presumptive challenges
to teach and to learn. Even a specific enquiry could be seen as an interroga¬
tive engagement between an enquirer and Mother Nature herself [Hintikka, 1989;
Hintikka, 1987], Suffice it to say that for our purposes, an argument can be said to
be dialectical when it is an interchange of speech acts under conditions of challenge
or test. Let us think of such arguments as dialectical in a generic sense. When
it is recalled that our word “dialectic” comes from the verb dialegestha, meaning
45 See also [van Eemeren and Grootendorst, 1992] and [Walton, 1989]. Cf. [Hansen, undated].
58
John Woods and Andrew Irvine
“argue” or “discuss,” then generically dialectical arguments are a natural fit and
certainly were recognized by Aristotle.
From this it is apparent that there are at least two senses of “dialectical argu¬
ment” and that they do not coincide. An argument from reputable premisses can
be transacted solo, on a lonely speech-writer’s lap-top late at night, but if it is
dialectical in the generic sense it cannot. For it to be true, as some modern com¬
mentators suggest, that a fallacy is an inherently dialectical structure, it must be
true that its false apparent goodness inheres in the conditions in virtue of which
the argument is an interchange of speech acts under conditions of challenge or
trial. Yet this would imply that the argument
All men are mortal
Some dogs are not men
Therefore, some dogs are not mortal
could not be a fallacy unless it arose in some actual give-and-take between real-life
disputants.
Some critics may not like our example of the lonely speech writer working
the solitary night shift in Ottawa or Hong Kong. They will say, and it will be
true, that in crafting the speech, the writer will be mindful of how it will play
in Parliament or at the next day’s news conference. He will attempt to marshal
premisses attractive to those whom the speech is meant to convince. This is very
often true, but it is also the sheerest nonsense that he could not be crafting an
argument which simply records the position of his beleaguered boss who is intent
on resigning honorably the next morning and intent only on declaring himself
honestly from premisses which others will not accept, and not caring whether
they do. Arguments composed solo are always the sorts of things that could be
transacted interpersonally, but that makes them no more intrinsically dialectical
in the generic sense than the fact that a boat can always be used for fishing makes
a corvette intrinsically a trawler. It is not even true that when spoken a solo
argument is dialectical in the present sense, for it might be spoken into a tape
recorder. It is, we suppose, perfectly open to the would-be theorist to constrain
the world “fallacy” in this generically dialectical way, but this is not Aristotle’s
way.
Necessitation, as we have said, is primitive in Aristotle’s logic. If we adopt
the idea that an argument is valid if and only if its premisses necessitate, or
imply, its conclusion, we put ourselves in a position to wonder whether there are
properties of validity that Aristotle might be brought to acknowledge, if only we
could ask him. It is clear that Aristotle understands the class of syllogisms to
be a restriction of the class of valid arguments. We have already said that it is
consistent with the constraints on syllogisms that validity itself could satisfy the
conditions that qualify it as having a Gentzen-logic. If Aristotle had views about
this possibility, he did not state them. But we are left, all the same, with two things
to theorize about: validity and syllogisity, which are linked, latter to former, by
the relation “is a restriction of.” It is hardly wayward, therefore, to think of there
Aristotle’s Early Logic
59
being two theories awaiting proclamation, a theory ©„ of validity, together with
its restriction, 0 S , a theory of syllogisity, and that it is only concerning the latter
that Aristotle offers an account.
Concerning 0„, we shall proceed conservatively. We adopt, on Aristotle’s behalf,
the common lexical stipulation that an argument is valid just when its premisses
necessitate, or entail, its conclusion. The task of 0 S is to give an account of
syllogisms. Validity abets the task. Aristotle requires a conception of validity that
enables syllogisms to be specifiable as a proper subset of protaseic arguments. A
methodological principle of interpretation now drops out:
Meth: Keep the account of validity as simple as is consistent with its
obligations in the theory of syllogisms.
We take Meth to commit us to what we might, with some looseness, call the re¬
ceived contemporary view of validity. It is nearly enough the conception of validity
fashioned in the metatheory of first-order logic, or for those whose tastes run that
way, in systems of strict implication such as S4 or S5 and their quantificational
extensions.
It might be wondered whether there is any need to be speaking of 0„ at all.
Why do we need a subtheory of validity if it has already been decided that Aris¬
totle’s validity is core Gentzen-validity? The answer is that a core Gentzen-logic
gives rise to different and incompatible full logics. If, for example, we restrict
Gentzen’s structural rules in such a way that deductions can have only single con¬
clusions (or only unit sets as conclusion), then such a logic is intuitionistic (a point
we return to later). Similarly, Gentzen’s structural rules (which define the core)
lay no constraints other than consistency on the theorist’s choice of operational
rules (which complete the specification of the full logic). So it is left open that a
given theorist, including Aristotle (if only we could consult him), might plump for
operational rules that make the full logic significantly non-classical.
In speaking, just now, of the subtheory, 0,,, we were speaking not of the core
logic but rather of a full logic of validity. There are several obvious questions to ask
about this full logic. One of them is whether it is classical. We have done nothing
so far to prove that it is classical. Perhaps no such attempt will succeed, given
that Aristotle himself has so little to say about validity. Perhaps the saner course
is to abandon any serious effort to pin a particular 0„ on Aristotle himself, but
it is still open to us to wonder whether some 0^ might be assumed on Aristotle’s
behalf. We ourselves think that this is a reasonable thing to try to do. For this we
want the comforts of Meth. It encourages the attribution, however tentative, of
what we think is the best full logic of validity. Others will disagree about “best,”
and as we proceed it is possible that we will unearth textual reasons for revising
the classical attribution which Meth defensibly calls for. Both these points are
taken up in what follows, but as a point of departure, we allow Meth to tell us
that 0„ is a classical theory of validity. 46
46 Terminological Note: We follow the convention by which classical logic is, of course, the
dominant part of modern logic. Who says that logicians have no sense of humour?
60
John Woods and Andrew Irvine
There is a further reason to want Meth. If we examine the case for saying that
Aristotle’s validity is (at least) core Gentzen-validity, we see that it depends upon
the assumption that syllogisms are non-vacuous, or proper restrictions of valid
arguments. But we have not shown that this is Aristotle’s express view. Were it not
Aristotle’s view, or were it not attributable to him on the basis of considerations
which are at least textually based, our case would be badly damaged. We want
some reason, in the absence of textual evidence to the contrary, to persist in our
attribution of a core Gentzen-logic; and this is what Meth gives us.
Meth bids us to attribute to Aristotle such a conception of validity so long as
doing so does not impede the development of © s , and provided also that the at¬
tribution is not contradicted or contra-indicated by Aristotle’s text. Our three
conditions warm us to this task in a particular way. If we propose as a plausible
assumption that Aristotle intended conditions Min, Non-Circ, and Prem+ to be
independent of one another, and of the necessitation condition, it follows immedi¬
ately that validity is not syllogisity and that it cannot be that an argument is valid
if and only if it satisfies conditions Min, Non-Circ and Prem+. We thus detach
ourselves from the view that “Aristotle’s conception of ‘following necessarily’ is
very different from the classical one.” 47
How plausible is the independence assumption? Suppose it were not true. Then
necessitation (or entailment) would satisfy all or some of the other conditions. If
it satisfied some but not others, this would leave the question of why. About
this Aristotle has nothing whatsoever to say. However, if entailment itself were re¬
quired to satisfy all the remaining conditions, there would be no difference between
entailment and, as we might say, syllogistic entailment, or between validity and
syllogistic validity. This is possible, but it cannot be Aristotle’s view. In as much
as syllogisms in the broad sense are only a proper subset of logically correct deduc¬
tions which Aristotle clearly recognizes to be so (e.g., impossibility proofs, ecthetic
proofs, conjunctive modus ponens arguments, immediate inferences), it must be
said that Aristotle acknowledges a conception of entailment other than syllogistic
entailment. It does not follow that non-syllogistic entailment is entailment in the
modern sense (although it is just this that Meth suggests).
If we go with Meth, the independence of Min, Non-Circ and Prem+ from the
necessitation condition is more directly established. It suffices to specify a valid
argument that fails all three conditions. Such an argument exists:
All A are A
Therefore, all A are A.
Valid by modern lights, this argument’s conclusion repeats a premiss, of which
47 Normore supports this claim by observing that Aristotle would not accept as a syllogism the
argument {“No animal is a human,” “Every human is an animal”} It- “No human is a human”,
presumably “because it does not contain three terms” [Normore, 1993, p. 447-448]. This is a
correct assessment by the formal lights of the Prior Analytics- but there is a chronologically and
conceptually prior reason to complain. It is that the argument contains inconsistent premisses,
which for Aristotle also rules out its syllogisity. It does not follow, of course, that the argument
is made invalid by a condition that denies it the status of syllogism.
Aristotle’s Early Logic
61
there is only one, and being a logical truth, its conclusion also follows from a
proper subset of its premisses, namely, the empty set.
This leaves the question of whether Min , Non-Circ and Prem+ are independent
of one another in their application to valid arguments. We proceed by constructing
a satisfaction-failure matrix for Min, Non-Circ and Prem+. The plus sign “+” in
the matrix denotes satisfaction of the condition, and
its failure. Given that satisfaction-failure is a bimodal pair, the number of to¬
tal satisfaction-failure combinations with respect to the three-membered set Min,
Non-Circ, Prem-h is 2 3 = 8 . The full matrix is given in Table 1.
Table 1.
Min
Non-Circ
Prem-h
Row 1
+
+
+
Row 2
-
+
+
Row 3
+
-
+
Row 4
-
-
+
Row 5
+
+
-
Row 6
-
+
-
Row 7
+
-
-
Row 8
-
-
-
Now recall that we are seeking an answer to the question whether conditions
Min, Non-Circ and Prem-h are independent. To show that they are, we need to
find a valid argument that satisfies each condition, but that also fails to satisfy the
other two, or a valid argument that fails to satisfy each condition, even though it
satisfies the other two. In other words, we need to find a series of valid arguments
corresponding to rows four, six and seven, and rows two, three and five. We
attempt to do so as follows:
Row 4- What is required for this row is the specification of a valid argument
which satisfies Prem-h but fails Min and Non-Circ. There is such an argument,
viz.,
All men are mortal
All ducks are quackers
Therefore, all men are mortal.
It is multi-premissed and hence satisfies Prem-h. But its conclusion repeats a
premiss and its second premiss is superfluous. Hence Non-Circ and Min fail.
Row 6: Here we require a valid argument whose conclusion does not repeat a
premiss, but which both has a superfluous premiss and does not have more than
one premiss. It is plain that such an argument does not exist, unless we allow as
valid those arguments whose conclusions are logical truths, and hence follow from
the empty set of premisses. Suppose that £ is an argument of this kind. Add to
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John Woods and Andrew Irvine
£ any premiss other than its conclusion. We call the resulting argument £*. Now
£* satisfies Non-Circ and fails Min and Prem+. It fails Min because £* is valid
without its premiss, and it fails Prem+ because it has only one premiss.
The issue of logical truths as the possible conclusions of syllogisms is an inter¬
esting one to which we shall return shortly. For the present we simply remark that
the independence question motivates our attention with respect to the status of
logical truths as conclusions of the empty set of premisses.
Row 7: What is needed here is a valid argument free of superfluous premisses
but which repeats a premiss as conclusion and has only one premiss. There are
such arguments, for example, the argument A lb A. Let £ be a valid argument
without superfluous premisses. If £’s conclusion both repeated a premiss and
had more than one premiss, £ would have a valid proper sub-argument, which is
contrary to the original assumption.
Row 2: Here we require a valid argument that satisfies both Non-Circ and
Prem+ but that fails to satisfy Min. Such arguments exist, and can be eas¬
ily constructed simply by adding a redundant premiss to any valid, non-circular,
multi-premissed argument, for example as follows:
All men are mortal
All Greeks are men
All Romans are Europeans
Therefore, all Greeks are mortal.
Row 5: In this case, we need to find a valid argument that satisfies both Min
and Non-Circ but that fails to satisfy Prem-h, and again, such arguments are easy
to find:
All men are mortal
Therefore, some men are mortal.
Row 3 : Finally, we come to Row 3, the case in which we need to find a valid
argument that satisfies both Min and Prem-h, but that fails to satisfy Non-Circ.
However, such arguments cannot be constructed. The reason is that any argu¬
ment that fails to satisfy Non-Circ will be one in which a premiss is repeated as
conclusion.
At the same time, if the premiss is repeated as the conclusion, and if the argu¬
ment contains premisses in addition to that premiss, it follows that those additional
premisses will be reduandant and that Min will not be satisfied, contrary to our
requirement.
What this shows is that although the set of Min , Non-Circ and Prem+ is inde¬
pendent of the validity condition, Min , Non-Circ and Prem-h are not independent
of each other.
Why should we be interested in whether a syllogism’s defining conditions are
independent? To answer this question, let us call a definition clean if its defining
conditions are independent of one another, and muddy otherwise. Muddiness is
evidently a matter of degree. By the proofs just above, we see that the proferred
definition of syllogisity is somewhat muddy. Why should we care about this? In
Aristotle’s Early Logic
63
a quite general way, muddy definitions obscure the net impact of their defining
conditions. A muddy definition presents us with two tasks instead of one. The first
task is the more important one. It is the task of producing conditions necessary and
sufficient for the definiendum. Though less important, the second task imposed by
a muddy definition is no mere call upon the theorist’s discretion. It is the task of
elucidating the interconnections among the defining conditions in virtue of which
indepedence is lost.
If simplicity were allowed to rule in such cases, we should be prepared to con¬
sider dropping a condition in favour of the independency of those that remain.
Supose that we dropped Prem+. Then, as it turns out, the other two are indeed
independent. What is more, this independence requires that validity be classical
enough to permit valid arguments from the empty set of premisses. But since this
is what Meth already bids us say about validity, the independence of Min and
Non-Circ is an attractive bonus.
The independence of Min and Non-Circ is exhibited by the satisfaction-failure
matrix in Table 2:
Table 2.
Min
Non- Circ
Row 1
+
+
R.ow 2
-
+
Row 3
+
-
Row 4
-
-
Independence is established by specification of valid arguments, one or more,
displaying the satisfaction-failure profiles of rows two and three.
Row 2 : It suffices to find a valid argument in which the conclusion repeats no
premiss but which contains a superfluous premiss. Such an argument exists:
All men are mortal
All ducks are quackers
Therefore, some men are mortal.
Row 3: It suffices to specify a valid argument which contains no superfluous
premiss but whose conclusion repeats a premiss. Such an argument exists:
All men are Greeks
Therefore, all men are Greeks.
We have it then, that Min and Non-Circ are independent conditions. Indepen¬
dence is lost by the addition of Prem+. So why add it? One reason for doing so
is that Prem+ is a condition expressly proclaimed in the Prior Analytics. This
leaves us with two options to think about. Option one: Withhold Prem-f- as a
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John Woods and Andrew Irvine
condition on syllogisms in the broad sense and reserve it for syllogisms in the
narrow sense. By these lights, it may be that Aristotle changed his mind about
syllogisms and came to apply Prem+ to facilitate the reductive programme in the
Prior Analytics, even at the cost of definitional muddiness. Option two: Impose
Prem+ as a condition on all direct syllogisms, broad or narrow, and deal with the
ensuing muddiness. This matter is discussed in greater detail in [Woods, 2001]. 48
7 INFERENTIALIZING THE CONSEQUENCE RELATION
It is interesting to note similarities between Aristotle’s account of syllogisms and
Bolzano’s logic of deducibility (Ableitbarkeit) , which in turn is widely seen as a
precursor of Tarski’s account of logical consequence [Corcoran, 1975; Thompson,
1981].
Bolzano requires deductions to have mutually consistent premisses [Bolzano,
1973]. Thus where, for Tarski, every sentence is a logical consequence of an incon¬
sistent set of sentences, for Bolzano inconsistent premisses bear the Ableitbarkeit-
relation to nothing whatsoever. Bolzano also requires that if a sentence is deducible
from a given set of sentences, it not also be deducible from any proper subset of
that set. Tarski, on the other hand, imposes no such constraint. Bolzano’s condi¬
tion makes his deducibility relation nonmonotonic. Tarksi’s consequence relation,
of course, is monotonic.
Bolzano’s two constraints on deducibility have exact counterparts in Aristotle’s
logic of the syllogism. Bolzano’s requirement that no proper subset of a set proving
a sentence prove that same sentence is simply Aristotle’s condition Min, adjusted
from a condition on syllogisity to a condition on deducibility. Bolzano’s require¬
ment that there be no deductions from inconsistent premisses can also be found in
Aristotle. It is directly provable from Aristotle’s condition Non-Circ with the aid
of the principle of (argumental) conversion, which is one of Aristotle’s common
rules of logic. Let
A
_B _
Therefore, C
be any syllogism. Then, by argumental conversion, the following is also a syllogism:
48 It is noteworthy that On Sophistical Refutations contains several examples of fallacies as
single-premiss arguments that are valid without premissory supplementation, i.e. examples which
are not enthymemes. David Hitchcock points out that this form of argument is prominently on
display in Aristotle’s fifteen examples of what he would later call the secundum quid fallacy. See,
for example, 166 6 , 37; 167°, 1, 7-9; 168 6 , 11; 180“, 23-24, 31-32, 33-34, 34-35, 35-36; 180 fc ,
9-10, 11-12, 14-16, 18-19, 20-21, 21-23. Other one-premiss arguments instantiate the fallacies
of equivocation (165 b , 31-32), illicit conversion of an A proposition (168 b , 35-169“, 3), and so
on. As Hitchcock says, “... twenty-six of Aristotle’s sixty-five fully detailed examples consist
wholly or partly of one-premiss arguments” [Hitchcock, 2000a, p. 214].
Aristotle’s Early Logic
65
A
—>c
Therefore, ->B.
Thus the rule of argumental conversion is syllogisity-preserving and non-syllogisity-
preserving. Consider now the non-syllogism
A
_B _
Therefore, A.
This argument fails to be a syllogism because it violates Non-Circ, the rule that
forbids circular syllogisms. Applying the rule of argumental conversion to this
non-syllogism gives us
A
—>A
Therefore, -u B
an argument in which the premiss set is expressly inconsistent. Since argumental
conversion is non-syllogisty-preserving, no argument of this form is a syllogism.
The minimality requirement makes for the nonmonotonicity of
Aristotle’s syllogisity just as it does for Bolzano’s Ableitbarkeit. Non¬
monotonicity is not a nineteenth-century discovery, still less one of twentieth-
century computer science. It was imposed at the very beginning of logic, by the
discipline’s founder. In making for the nonmonotonicity of syllogisity, Min also
endows syllogisms with two other modern-looking features. A logic is linear when
each premiss is used exactly once. It is easy to see that a linear logic is thus also
a relevant logic for that sense of relevance in which something follows relevantly
from a set of premisses if there exists a deduction of that sentence in which all
those premisses are used. We have it, then, that although Min imposes additional
constraints as well, in imposing Min Aristotle is producing the first linear, hence
relevant, logic in this subject’s long history.
The consistent-premisses condition is also consequential. It expressly provides
for an implication relation (syllogistic implication, as we might say) which fails
the condition ex falso quodlibet, according to which everything is deducible from
an inconsistent set of sentences. Aristotle constrains syllogisms in such a way that
nothing syllogistically follows from an inconsistency. In so doing Aristotle provides
that the logic of syllogisms is the first paraconsistent logic.
The requirement that syllogisms be constructed solely of propositions also bears
thinking about, quite apart from the audacity of the claim that anything stateable
is stateable without relevant loss in the language of propositions. Recall that a
proposition in Aristotle’s sense is a statement in which one thing is said of one
thing. With the exception of negation, they are also statements free of connectives.
Syllogisms are sequences of propositions. Each line of a syllogism is occupied by
one and only one proposition. Were it otherwise, were it the case that a line was
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John Woods and Andrew Irvine
occupied by more than one proposition, then it would be a line in which more than
one thing is said of more than one thing. To see how the propositional constraint
works as a construction rule for syllogisms, it is essential that we see syllogisms as
sequences in which at each line one thing (only) is said of one thing (only). This
being so, syllogisms cannot have multiple conclusions. It might here be noted that
any logic in which dilution fails and in which the standard operational rules are
upheld ( e.g ., the introduction rules) is an intuitionistic logic. To generate from
such conditions a classical logic, multiple conclusions must be admitted . 49 Thus
the first logic was at least in the spirit of an intuitionistic logic.
The theory of syllogisms was the first linear (hence relevant and nonmonotonic),
paraconsistent and intuitionistic-like logic ever known. Syllogisms are classically
valid arguments constrained in ways that make them very different from the clas¬
sical validities that they would be if left unconstrained. There are worlds of dif¬
ference between consequence relations that do (and do not) permit closure under
the proper subset relation on premiss sets, that do (and do not) permit multiple
conclusions. The very fact that Aristotle did not labour to bring forth a theory of
unadorned validity indicates that, while essential to his purposes, validity deliv¬
ers none of the special goods for which these truly encumbering constraints were
needed. As we have seen, corresponding to any syllogism is its conditional state¬
ment, and corresponding to it is a rule of inference (so-called). Aristotle wanted a
logic whose rules of inference—or at least the syllogistic rules—were not given the
free flight of classical rules.
Why did Aristotle think it so important to constrain his rules? The answer
appears to be that he wanted the rules of his logic of syllogisms to be usable
in systematic accounts of real-life argument and thinking. Left without all this
baggage, the validity rules regulate a content-free notion of consequence, which is
defined simply for sequences of linguistic, truth-valuable entities, and whose pri¬
mary function is truth preservation. But no rules that deliver on these objectives
and none other are at all realistic as rules of real-life thinking and debate. It is
one thing to say that inconsistent statements imply any statement; it is another
thing entirely to say that when real-life reasoners are faced (in real time) with an
inconsistency in their belief sets or commitment sets they do (or should) accept
or commit to every statement whatever. Similarly, it is one thing to say that any
truth preserving argument remains truth-preserving under arbitrary supplemen¬
tation of premisses arbitrarily many times; but it is another thing altogether to
say that when one has a truth-preserving argument at hand, it is always good
argumentational strategy to make any supplementation of it that preserves truth.
Such would be the course of risk aversion taken to ludicrous extremes.
Aristotle is essaying a bold experiment. He is taking seriously the idea that
usable real-life rules for the conduct of argument and thinking can be got from
context-free truth conditions on a purely propositional relation, provided the right
constraints are imposed. In their unconstrained form, whether one proposition
logically implies another tells us virtually nothing about whether it would be ap-
49 [Shoesmith and Smiley, 1978, p. 4]
Aristotle’s Early Logic
67
propriate, helpful, realistic or possible to conform one’s argumentative or cognitive
strategies to that bare fact of logical consequence. Aristotle’s gamble is that facts
about logical consequence do give the requisite guidance for argument and reason¬
ing when constrained in the right ways.
Aristotle’s example serves to remind us that the kind of logic a logician thinks
up is, unless he is simply being playful, more or less directly the product of what
the logic is wanted for. In a celebrated quip, W.V. Quine proposed that logic
is an ancient discipline, and that since 1879 it has been a great one. The year
1879 marks a momentous event in the history of logic: the publication of Frege’s
Begriffsschrifft. It takes no disparagement of Frege’s great accomplishment to
make the point that Quine has judged Aristotle and Frege on the wrong basis.
Frege needed a streamlined second-order predicate logic, coupled with something
resembling set theory, as the analytical home for arithmetic. Frege was a logicist.
He thought that it was possible to prove that the truths of arithmetic were analytic.
This he sought to do by finding an analytic discipline to which the truths of
arithmetic could be reduced without relevant loss. This host theory for arithmetic
was second-order logic plus set theory (nearly enough), to which Frege himself
made utterly seminal contributions. But it is absurd to abjure Aristotle’s logic
for failing Frege’s objectives. Given Frege’s purposes there was nothing to be
said for making the consequence relation linear, nonmonotonic, paraconsistent and
intuitionisitic-like. Doing so would not have advanced Frege’s logicist ambitions
one jot and would, in fact, have impeded their realization in various ways. In
contrast, Aristotle had very different objectives, none of them bearing on the
epistemology of arithmetic. Given the objectives that he had, the constraints
imposed by Aristotle on the consequence relation were very much in the right
direction. Aristotle’s project, then, was to “inferentialize” truth conditions on
a consequence relation purpose built for service in a realistic account of human
cognitive and argumentative practice. Much of the so-called nonstandard logic of
the present day is in various ways a continuation of this project to inferentialize the
consequence relation, to retrofit it for work that is psychologically real or some
approximation thereto. Opinion is divided as to whether the desired goods in
theories of cognition and argument can in fact be delivered by inferentializing the
consequence relation (see e.g., [Woods, 1994] and [Woods, 2003]). It is a question
for those nonstandard logics, no less than for Aristotle’s original logic.
8 ARISTOTLE’S VALIDITY
We have proposed that Aristotle holds what modern logicians call a classical con¬
ception of validity and, correspondingly, a classical conception of logical implica¬
tion or entailment. We make this proposal in the face of the fact that Aristotle
gives no account of these things anywhere in his writings. We have grounded our
proposal on two basic facts. One is that Aristotle’s logic does not require that
validity be non-classical and is in no discernible way improved by assuming so.
The other is that, if we assume that the syllogisity conditions are non-redundant,
68
John Woods and Andrew Irvine
then Aristotle’s validity must fail to be everything that syllogisity is required to be,
validity itself excepted. So there is good reason to believe that Aristotle’s validity
is not nonmonotonic, not relevant, not paraconsistent and not intuitionistic. Of
course, it does not strictly follow from these facts that validity is not non-classical;
but it does make the nonclassicality assumption highly implausible.
Against this is Aristotle’s Thesis, so-called by Storrs McCall. Aristotle advances
this thesis at Prior Analytics B4 57 6 , 4-7; so strictly speaking it does not fall within
the ambit of Aristotle’s earlier logic. Even so, Aristotle’s Thesis matters for certain
things we wish to say about this logic. In particular, it matters for the claim that
in his early writings Aristotle’s concept of validity is classical. Whether it is or
not is complicated by the fact that different scholars read the thesis in different
ways. A further complication is the uncertainty that attends the question as to
what Aristotle’s Thesis is a thesis about. Aristotle writes,
But it is impossible that the same thing should be necessitated by the
being and by the not-being of the same thing. I mean, for example,
that it is impossible that B should necessarily be great if A is white
and that B should necessarily be great if A is not white.
McCall takes this passage to assert that
->($ lb -1$)
he., that no proposition entails its own negation (see [McCall, 1996]). This is also
the interpretation of [Routley et al ., 1982, pp. 132, 343]. McCall also interprets
the passage as denying the validity of each of the arguments $ lb r ->4>” 1 and
r -,$i |t- <f>. Woods, on the other hand, reads the passage as asserting that of the
pair of arguments
$ _ _
Therefore, V I ; Therefore, $
at most one can be valid [Woods, 2001, pp. 55 ff]. Despite their important dif¬
ferences, these conflicting interpretations have common features of consequence
for our claim that Aristotle’s validity is classical. On the McCall-Routley inter¬
pretation it cannot be true either that a necessary truth is entailed by any set of
premisses or that an inconsistency entails any consequent whatever. On the Woods
interpretation, the same is true. Necessary propositions are not consequences of
arbitrary premisses; and if transposition holds true, neither is it the case that
inconsistencies entail any consequent whatever. Note, however, that these conse¬
quences follow only if Aristotle’s Thesis is a thesis about validity. Also required
is a strong interpretation of invalidity, which we shall call counter-validity. An
argument form is counter-valid when all its instantiations are invalid.
We will not here attempt to settle the question of how best to interpret Aris¬
totle’s words. It suffices for our purposes that if the thesis is understood to be
attributing counter-validity, then on either interpretation, whether that of McCall
Aristotle’s Early Logic
69
and Routley or of Woods, there will arise difficulties for the claim that Aristo¬
tle has a classical conception of validity. In what remains of this section we will
thus concentrate on showing that if Aristotle’s validity accepts just three classical
principles, viz., A-elimination, V-introduction and transitivity, it is easily shown
that Aristotle’s Thesis is false or, if true, that at least it cannot be a theory
about validity. Bearing in mind (we say) that the thesis is for no $ and H* is
it the case that both $ lb H* and r -iH> lh it" 1 are valid, it suffices to instantiate
$ and to opposite effect. Let r H> = C V ->C” 1 and r H r = C V ->C V D ~*, for
arbitrary C and D. Then $ lh Hi, by V-introduction. Since r -i$ n is ->C A C,
then r -i$ lh C, by A-elimination. But C lh C V ->C V D, by V-introduction.
Hence r -i$ lh C V ->C V D~* (be., Hi), by transitivity. So $ lh HI and r ->H> lh Hi’ 1 :
for some $ and H*, both $ lh Hi and r ->H> lh H'’ 1 are valid arguments. (A similarly
strong argument can be given in the case of the McCall/Routley interptation.)
Thus Aristotle’s Thesis is false if these three principles hold for validity. If
Aristotle’s Thesis is indeed incompatible with ex falso, the incompatibility is now
a technicality. No claim is over turned by false propositions incompatible with it.
Perhaps we should try to imagine whether Aristotle would himself have acquiesced
to our three rules. We think it exceedingly likely that he would have. But right or
wrong, it can also be shown that ex falso is true using only the following principles:
reflexivity, monotonicity, transitivity, “conversion,” and modus ponens. We do so
as follows: for all H>, Hq and x,
(1) H> lh H> is valid Reflexivity
(2) Hq $ lh $ is valid Monotonicity, 1
(3) -i$, H> lh ->Hi is valid Conversion, 2
(4) If H> lh H> is valid, so is -iH>, H> II—’Hi Transivitity, 1, 2, 3
(5) -'H>, H> II—'H* is valid Modus ponens, 1, 4.
Since Hi is arbitrary, it covers all negations r -'X~ l - Hence all x are validly
deducible from any { r ->H>
If the proof is good, it suffices to topple Aristotle’s Thesis if it is indeed incom¬
patible with ex falso (or more directly, with its dual, which sanctions the derivation
of logical truths from arbitrary premiss-sets, including the null set). 50
Aristotle himself endorses modus ponens and conversion, and he allows tran¬
sitivity for hypothetical reasoning and, apparently, for chains of syllogisms. 51 It
remains to wonder what he may have thought about reflexivity and monotonicity
considered not as conditions on syllogisity (in which case they both fail), but as
conditions on validity. It is well to note in passing our all but complete surrender
to Meth: given that the syllogisity conditions are independent of the validity con¬
dition, make it your point of departure concerning any property not a property
50 Also toppled is McCall’s Aristotle’s Thesis, as witness the validity of r -'('t>V -’$) It <t> V
The importance of this turns on whether we have independent reason to attribute to Aristotle
the derivation principles which sanction such arguments.
51 There is a problem with chains. Although championed in the Scholastic tradition and beyond,
it is hard to find passages in which chains of syllogisms are given syllogistic recognition in the
Prior Analytics. However, Topics 100“, 27 offers some encouragement.
70
John Woods and Andrew Irvine
of syllogisms but which could be a property of valid arguments to ascribe to it
validity until you have found good reason not to.
What, then, about reflexivity? If reflexivity holds in @„, we must say that
every statement necessitates itself. On one interpretation this is nonsense. It is
nonsense if self-necessitation is so understood that every statement makes itself
true. We take it without further ado that there is nothing to be said for the view
in which every statement is its own verification. What makes the self-necessitation
claim sound wrong is a misinterpretation of “necessitation.” Aristotle means by
r 4> necessitates <k n that it is guaranteed that 'k is the case if $ is. Reflexivity
or self-necessitation is just a special case of this: <f> is the case if $ is the case.
That this is so is encouraged by remarks at Posterior Analytics 73°, 4-6, where
Aristotle derides an imaginary opponent for complaining that all demonstrations
are circular. Aristotle claims that the complainant is saying “nothing but that
if A is the case A is the case,” and he adds not that this is untrue but, rather,
that “it is easy to prove everything in this way.” For our present purposes it is
enough that Aristotle does not here disallow reflexivity, but it is interesting to
note that his objection against the complainant is that the purported proof that
demonstrations are circular is an argument grossly in the form
(1) Demonstrations are circular
(2) Therefore, demonstrations are circular.
He adds, ironically, that of course it is easy to prove everything in this way. This
is irony twice over. Aristotle means that whereas (1) lb (2) is a valid argument, the
last thing it is, is a proof (for if it were, everything could be proved). Aristotle is
also lampooning his critic by so representing the critic’s own argument as to make
it a case of the thing he is objecting to. So we conclude that we lack sufficient
cause to make Aristotle’s validity irreflexive.
What of monotonicity? It is helpful to bear in mind the sort of thing that
condition Min is supposed to provide. In syllogisms, conclusions follow not only
from their premisses but also because of them. In introducing Min , Aristotle takes
pains to mark a contrast. It is a contrast between necessitation from and neces¬
sitation because of. It is important that Aristotle does not say that there are no
necessitations-from. In fact, every syllogism is a necessitation-from. A syllogism is
also something more; it is a necessitation-because-of. If the distinction is to have
a point, there must be properties of necessitations-because-of that necessitations-
from do not have. One such is the property of being causative of conclusions
of syllogisms. In mere necessitations-from there will be premisses that are not
causative of conclusions. These fairly enough can be said to be irrelevant to those
conclusions. But if necessitations-because-of banish irrelevant premisses, it can
only be expected that, in contrast, necessitations-from allow them. That is what
monotonicity allows, too. Let 4>i,..., lb be any valid argument with relevant
premisses. Let x be any statement irrelevant to 4' (and to all the for that
matter). Monotonicity nevertheless sanctions the validity of $i,...,$ n lb 4'. It
Aristotle’s Early Logic
71
sanctions what Aristotle himself appears also to sanction. So we conclude that
Aristotle would have no occasion to refuse the monotonicity principle for validity.
Against this it might be argued that monotonicity goes further than anything
portended by Aristotle’s distinction between necessitation-from and necessitation-
because-of, and that it is this additional feature that Aristotle might well have been
minded not to accept. Monotonicity expressly allows what the contrast between
“from” and “because of’ certainly does not expressly allow, viz. , that it is always
all right to supplement the premisses of a valid argument in such a way that
the resulting argument is valid and inconsistently premissed. 52 So let us turn to
inconsistency. Inconsistency is not much discussed by Aristotle. It is difficult to see
a stable policy on inconsistency and difficult therefore to see why the present point
should persuade us to make Aristotle’s validity nonmonotonic. We have already
argued that Aristotle’s validity is captured by the core Gentzen conditions, one of
which is monotonicity. It will take the heft of substantial evidence to shift us from
this view.
9 NECESSITIES
The theorems of the earlier logic and of the Prior Analytics register essential truths
about direct syllogisms. These are Aristotle’s “truths of logic.” Either these truths
of logic are themselves logical truths or they are not. If they are, the reasoning
which underwrites them cannot be the reasoning which they themselves describe.
If they are not, then presumably they are nonlogical necessary truths, and the same
conclusion follows. No truth about direct syllogisms is the conclusion of a direct
syllogism. As we saw earlier, this comes as no shock to Aristotle. Hypothetical
syllogisms, such as reductio per impossibile arguments and ecthetic proofs, are not
direct, and yet they are indispensable to the story that Aristotle wishes to tell
about those that are direct. Even so, it is somewhat unsettling that in direct
syllogisms no logical or necessary truths may appear as conclusions. This excludes
too many cases that would appear to be paradigms of perfect syllogisms, as witness
All squares are rectangles
All rectangles are four-sided
Therefore, all squares are four-sided.
The exclusion of syllogisms such as these is so implausible that, Meth aside, we
might consider rethinking our decision to attribute to Aristotle a classical notion
of validity.
52 Let it be noted that monotonicity does not give ex fatso. It provides only that whenever
there is a valid argument there is a valid superargument of it with inconsistent premisses, and
whose conclusion is the conclusion of the original. Ex fatso is stronger. It provides that an
inconsistent set of premisses endorses everything as conclusion. An equivalent difference is this.
Let v l' be valid from inconsistent premisses by monotonicity alone; then it follows that 'h is valid
from a proper subset of those premisses. On the other hand if, by ex fatso alone, 'I' is valid from
inconsistent premisses it does not follow that >1' is valid from a proper subset of them.
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John Woods and Andrew Irvine
Necessary truths are a disaster for syllogisms. They are a disaster, that is, if
counter -ex falso is true. We have said why counter-ex falso appears to be true
and we have said why it is likely that Aristotle himself would have acknowledged
its truth. If this is right, then the disaster that necessary truths produce is that,
for reasons that Aristotle would not dispute, they wreck the project of laying a
deductive substructure for the sciences. It is true that purely formal necessities
such as “All A are A” are spared the embarrassment of failing the Mm condition
on syllogisms, but they are spared only by being victimized by a different em¬
barrassment; for they cannot even be expressed, never mind proved, in Aristotle’s
theory of syllogisms.
It might be said that no logical theory has ever had success in dealing with
nonlogical semantic necessities such as “All red things are coloured.” So why should
we impose on Aristotle’s theory an expectation that no one else has met? There is
little to recommend this leniency. It overlooks semantic necessities of precisely the
sort that an Aristotelian apparatus is designed to capture, namely, statements such
as “All bachelors are unmarried” and “All squares are rectangles” that are true
by definition. What makes “All red things are coloured” a problem for logicians is
that “red” appears not to have a definition. Its problem, at least in part, is that
“All red things are coloured” is necessarily true, but not a truth of logic and not
a definitional truth. 53
A related difficulty attaches to semantically valid but formally invalid arguments
such as
All Granny Smiths are apples
All apples are red
Therefore, all Granny Smiths are coloured.
Formally invalid, this argument commits the fallacy of four terms. Yet it is also
true that given the meanings of “red” and “coloured,” the argument cannot have
a false conclusion if its premisses are true. Perhaps we could remedy the situation
by relaxing the prohibition on extra terms. We might say that an argument
Q (A, B)
Q (g, C) _
Therefore, Q ( A , D )
commits the fallacy of four terms except where the Qs are, in this order:
All-All-All, or Some-All-Some, or All-All-Some and C semantically entails D; or
All-No-No, or No-All-No, or Some-No-Some ... not, and C is semantically entailed
by D.
But even if this is a complete rule, covering all the right cases, it still leaves
the fact that there is no rule that tells us how to determine in the general case
whether n semantically entails 7r' or is semantically entailed by it.
Whatever we decide to say about semantic necessities, the necessary truths of
mathematics cry out for rescue. In such an extremity it is permissible to clutch
53 See [Searle, 1959]; cf. [Woods, 1967] and [Woods, 1974].
Aristotle’s Early Logic
73
at straws. Aristotle recognizes different grades of necessity. Corcoran says that in
Aristotle’s modal logic there are up to five different and apparently incompatible
systems competing for theoretical disclosure. 54 A straw presents itself: suppose
that we granted to mathematical truths an attenuated necessity, a necessity of
lesser grade than such full-blown semantic truths as “All bachelors are unmar¬
ried” and “Everything red is coloured.” To this end, we could appropriate the
expression “mathematically necessary” and contrast it with semantic necessity by
stipulating that it is a grade (perhaps the strongest grade) of non-formal necessity
for which counter-ex falso fails. In saying so, we would be adopting for the truths
of mathematics a kind of nomological necessity that lacks full generality. Scientific
laws are sometimes held to be nomologically necessary. By this is meant that they
are unfalsifiable in their own domains, i.e., their necessity is discipline specific; but
they fail, perhaps vacuously, in other domains. This suggests that mathematical
necessities might be reconciled to this conception. It cannot be said that their not
being true is in no sense possible, but rather that their falsehood is in no sense
possible within the domain of plane figures, or natural numbers, or topological
spaces, or whatever else. In contrast, we might expect to find full-blown semantic
necessities to be true in all domains. Unlike “Every triangle has the 2R property,”
whose failure is not possible in plane geometry, we would have “Every bachelor is
unmarried” whose failure is not possible in any domain.
Yet this will still not work. If “Every triangle has the 2R property” fails out¬
side of geometry, say in metallurgy, it does so vacuously. It is not a statement
formulable in metallurgy, so its negation also fails. Another way of saying this is
that the terms “triangle” and “has the 2R property” carry no metallurgical refer¬
ence; they are empty terms in metallurgy. It is the same way with necessities of
full-blown purport. “Every bachelor is unmarried” also fails in metallurgy, since
“bachelor” and “unmarried” are empty terms there. Aristotle requires terms to be
non-empty—and this is the source of his infamous doctrine of existential import.
There is little doubt that Aristotle would welcome the suggestion that mathemat¬
ical necessities fail in domains in which their embedded terms are empty. Admit¬
tedly, saying so leaves open the problem of how the quantitative sciences are to
be understood; but Aristotle has this problem anyway. It instantiates the gen¬
eral prohibition of statements from one discipline serving as premisses in another
discipline. So the present suggestion does not create a new problem for Aristo¬
tle; it simply exemplifies a problem that was already there. So, for good or ill,
Aristotle would welcome the suggestion that “Every triangle has the 2R property”
fails in metallurgy; but he would not welcome, nor should he, the suggestion that
full-blown necessities fare any differently.
It seems best to give up the notion of full-blown semantic necessities; that is,
nonformal truths of a grade of necessity strong enough to satisfy counter-ex falso.
On this suggestion, we evade the problematic provisions of counter-ex falso by
pleading that there is no grade of necessity attaching to nonformal truths suffi-
54 See again [Corcoran, 1974c, p. 202]. This is also McCall’s view. See [McCall, 1963] and
[Patterson, 1995].
74
John Woods and Andrew Irvine
ciently strong to trigger the metatheorem. For let $ be any nonformal necessity.
Then its failure is guaranteed in all alien domains. Thus there will be some state¬
ments from which >P does not follow. These will be statements true or
false in any domain in which $ fails on account of alienation, and these will be
precisely the domains in which the necessity of l P fails to satisfy counter-ez falso.
Still, this is too much to hope for. In the shady glades of watered-down ne¬
cessity, one hand washes the other. Our current speculation provides that if
is a nonformal necessary truth, there will be a domain in which its negation is
impossible and which, in alien domains, it is neither possible nor impossible. (In
fact, it turns out not to be a proposition there.) Correspondingly, r -'\P”' will have
a necessary negation in the home discipline and in all others it will be neither
possible nor impossible. Consider, then, the argument
Therefore, *P
in which, for some discipline D, \P is a necessary truth. Then r< Pi A $2 A -i\P’ 1 will
be impossible in D. Beyond D, the argument vanishes, owing to what might now
be called reference failure. So we will say that our argument is D-valid, and if it
meets the other conditions on syllogisity, it is a D-syllogism. Let us suppose that
it does meet these conditions. Let % D be the empty set of premisses from D. Then
Therefore, $
is a D-valid argument since is impossible in D and is already conjoined
with the putative membership of 0°. But 0° lb IP is a D-valid sub-argument of
$i,$ 2 lb VB, which is itself D-valid. Hence $i ,$2 lb $ is not a syllogism, contrary
to our hypothesis.
We have not found a way of attenuating mathematical and other nonformal
necessities in ways that avert the problem they pose for syllogisms. The disas¬
ter they occasion recurs. Aristotle thinks that every science is (or contains) the
demonstrative closure of first principles, that first principles are necessary, and
that their necessity is preserved in their closures. 55 Either those necessities are
full-blown, i.e., they hold in every domain including the null domain, or they are
domain- or discipline-relative. That is, they hold in their own domains but fail for
want of reference in every other, including the null domain. If the first possibility
holds, there will be no sciences since there will be no demonstrative syllogisms.
In every putative science for which there are full-blown necessary truths, they will
follow from the null set of premisses. If the second possibility holds, the same
unwelcome result awaits: once again there will be no sciences since there will be
no scientific syllogisms. For, again, let $ be a proposition necessary in D and
only in D. Then $ follows from 0 D , the empty set of premisses in D. 0-° lb IP is a
58 Thus to have scientific knowledge of something is to know the cause or reason why it must
be as it is and why it cannot be otherwise (Posterior Analytics 71 8 , 17-33).
Aristotle’s Early Logic
75
valid proper sub-argument of any D-valid premissed argument for U/. Yet no such
argument is a syllogism. It follows that there are no demonstrations in D. So D is
not a science.
We have been taking Min to preclude syllogisms with valid sub-arguments. As
things have developed, Min and counter-ea; falso collide with one another mo¬
mentously. For together they make science impossible. This is a consequence
sufficiently disagreeable to call Non-Circ and counter- ex {also both into question.
There are plenty of logicians, e.g., those of the Anderson-Belnap persuasion, who
would think that counter- ex falso is the obvious choice for rejection. Counting for
this, in an indirect sort of way, is that there is something that obviously counts
against the rejection of Min. What counts against rejecting it is that Aristotle
seems expressly to proclaim it. His commitment, if such exists at all, to counter¬
ed falso is nothing that Aristotle ever expressed; and given the seriousness of its
conflict with a principle Aristotle does express, there appears to be nothing to be
said for its retention.
We lack the space here to reflect further on how best to adjudicate the tension
between Min and counter-ex falso , since doing so is highly conjuctural. However,
interested readers may consult [Woods, 2001, ch. 7].
10 REFUTATIONS
An important precursor of the Aristotelian refutation argument is the eristic ar¬
gument, prominently on display in Plato’s Euthydemus. Eristic argument, in turn,
is a refinement of the Socratic elenchus, found in such dialogues as the Euthyphro,
Laches, Charmides and Lysis. Bonitz identifies twenty-one different eristic argu¬
ments in the Euthydemus . 56 Each begins with the assertion of a thesis. A second
party (either Euthydemus or Dionysodorus, depending upon the particular case)
presses the first with questions. Most of these are Yes-No questions, to which
the expected answer is nearly always in the affirmative. In some cases, the ques¬
tions have an Either-Or structure, and the answerer responds by picking one of
the disjuncts. 57 The questioner attempts to draw conclusions from the answerer’s
responses. Usually this is done deductively. “The refutation is successful when the
questioner is able to draw from his interlocutor’s admissions either some conclu¬
sion incompatible with the original thesis (not necessarily its direct contractory)
or some absurdity whose derivation used the thesis as a premiss.” 58
The form of the interplay between Euthydemus and his brother
Dionysodorus is very similar to that of the Socratic elenchus, except that tougher
constraints are imposed on what the answerer is permitted to say. Thus an eristic
argument is an elenchus with stiffer rules.
56 See [Bonitz, 1968]
57 In argument 19 there are three examples of information-solicitation questions, e.g., “Can you
name three types of craftsmen by the work that they do?” Again, see [Bonitz, 1968].
58 [Hitchcock, 2000b, p. 60].
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John Woods and Andrew Irvine
Eristic arguments are not problem free, something that Aristotle would attend
to in his On Sophistical Refutations. As Hitchcock says,
The most probable origin of professional eristic ... is Socrates himself.
This is not to say that the brothers got their repertoire of fallacious
tricks from Socrates, but that they practised the type of refutation in
which Socrates engaged, and inserted into it the trickery which subse¬
quently earned the name ‘sophistry’ [Hitchcock, 2000b, p. 63].
The distinction between syllogisms-as-such and syllogisms-in-use affects Aris¬
totle’s conception of refutation in an interesting way. The distinction is exem¬
plified by those arguments (syllogisms-as-such) that are the core of refutations
(syllogisms-in-use). How, then, do refutations work? As with eristic arguments,
Aristotle provides that there are two participants, Q, a questioner, and A, an an¬
swerer. A proposes a thesis T. 59 Q’s role is to question ( erotan ) A , putting ques¬
tions to him, answers to which are formatted as simple (non-compound) declarative
sentences, or propositions ( protaseis ) in Aristotle’s technical sense of the term. A’s
answers are thus available to Q as premisses of a syllogism, (let’s call it Ref), which
it is Of s role to construct. If Ref is constructed and if its conclusion is the con¬
tradictory, ->T, of A’s original thesis, then Q’s argument is a refutation of T. In
this we see the pure form of Locke’s ad hominem, for Refs premisses are A’s own
principles and concessions (and nothing else); and Refs conclusion, the contradic¬
tory of A’s thesis, is got by pressing those concessions with their consequences.
For concreteness, we now imagine a simple case of refutation. A’s thesis is T. Q’s
refutation is
C
_B _
Therefore, ->T
in which C and B are A’s concessions and Tis syllogistically derived from these. It
is a noteworthy feature of our case—a feature which generalizes to all refutations—
that A’s thesis T cannot be a premiss of Ref. Here is why. Suppose that
T
A
-i T
were a valid argument. As it stands, it has the appearance of a reductio, provided
that we understand T as a hypothesis rather than as a premiss. But reductios are
not refutations. If our present argument is to make the grade as a refutation, it
must be a syllogism. Since it is a valid argument, the set of its premisses, together
with the negation of its conclusion, is inconsistent. This is the set {T, A, T}, which
is the same set as the set of the argument’s premisses alone. But syllogisms cannot
59 A word of caution: in Aristotle’s usage a thesis is a paradoxical claim. This is not here its
intended sense. Aristotle’s word is problema.
Aristotle’s Early Logic
77
have inconsistent premiss sets. Hence our argument is not a syllogism, and not a
refutation.
Since Ref is a syllogism, {C, B , T} is an inconsistent set. Hence at least one
sentence of the three is false. Because Ref is a refutation of T, it is attractive to
suppose that Ref establishes that it is T that is false (for what else would Ref s
refutation of T consist in but showing that T is false?) Yet this will not do. Saying
so is fallacious. It is the fallacy of distributing negation through conjunction, said
by some to be Aristotle’s fallacy of Noncause as Cause. (It is not, but let that
pass; see [Woods and Hansen, 2003].) What we require is some principled reason
to pick out T, rather than C or B, as the unique proposition refuted by Ref. How
are we to do this? 60 There are two possible answers to consider. The first will
prove attractive to people who favour a broadly dialectical conception of fallacies.
The second will commend itself to those who think of fallacies as having a rather
more logical character. We examine these two possibilities in turn.
First Answer (Dialectical): The first answer proposes that question-and-answer
games of the sort played by Q and A are subject to the following pair of linked
dialectical rules:
Premiss-Selection Rule: In any dispute between Q and A in which
Q constructs a refutation of A’s thesis T, Q may use as a premiss of
his refutation any affirmation of A provided that it is subject to the
no-retraction rule.
No-Retraction Rule: In any such dispute as above, no answer given by
A to a question of Q may be given up by A.
It is entirely straightforward that among A’s affirmations, germane to his de¬
fence of T, T itself is uniquely placed in not being subject to the No-Retraction
rule. If it were, then once affirmed it could not be retracted. But if it couldn’t
be retracted, it couldn’t be refuted (or, more carefully, couldn’t be given up for
having been refuted). This would leave refutations oddly positioned. The rules
of the game being what they are, a refuted proposition would be precisely what
the refutation could not make (or allow) A to abandon. Thus it may be said that,
according to our first answer, if our refutation game is a procedurally coherent
enterprize, then, of all of A’s relevant affirmations, T alone stands out. It is the
only affirmation that A can retract and it thus qualifies for the proposition which
Q’s refutation, Ref, refutes.
It may be thought, even so, that it is unrealistically harsh to hold all of A’s
other affirmations to the satisfaction of the No-Retraction rule. A little reflection
will discourage the complaint. In the give-and-take of real life argument some
latitude is given to A. He is allowed to retract some affirmations some of the time.
There are limits to such sufferance. If there were not, then any thesis would be
made immune to any refutation of it. If A always had the option of cancelling one
60 The same problem visits the idea of counterevidence to a scientific theory construed holisti-
cally.
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John Woods and Andrew Irvine
of Q’s premisses, rather than giving up his own T, then T would be made strictly
irrefutable by any Q whose opponent were prepared to exercise the option. Thus
the No-Retraction rule can be considered an idealization of this limit.
Second Answer (Logical): According to our second answer, that Tis the uniquely
positioned proposition that Q’s Ref refutes can be explained with greater economy
as follows. Looking again at Ref ,
C
_B _
Therefore, ->T
we see that T is distinguished as the proposition refuted by Ref precisely because
it satisfies a certain condition. Before stating the condition, it is necessary to
introduce a further fact about syllogisms. Aristotle requires that syllogisms always
have consistent premiss sets. For suppose that they did not, and let
A
-i A
Therefore, B
be a syllogism. Then since syllogisms obey the argument-conversion operation,
and since conversion preserves syllogisity, our imaginary syllogism converts to
A
—iB
Therefore, A
in which the conclusion repeats a premiss. This is explicitly forbidden by Aristo¬
tle’s definition:
A syllogismos rests on certain statements such that they involve neces¬
sarily the assertion of something other than what has been stated ( On
Sophistical Refutations, 1, 165 a ,l-3).
Whereupon, since the second argument is not a syllogism and yet is an argument
converse of the first, neither is the first a syllogism. (This ends our current aside.)
The condition proposed by the second answer to our question is now given as
follows:
Cl: Since Ref is a syllogism, its premiss set is consistent. Let Aff
be the set of A’s affirmations with respect to Ref. Thus Aff is the
set {T, C, B}. Since Ref is valid Aff is inconsistent. It is easy to
see that Aff possesses exactly three maximal consistent subsets: { C,
B}, {C , T), {B, T}. We will say that a maximal consistent subset of
Aff is excluded by Ref if and only if it does not syllogistically imply
Refs conclusion, ->T. Thus { C, T } and {B, T } are excluded by Ref,
and {T} is their intersection. Thus T is the proposition refuted by
Ref precisely because it is the sole member of the intersection of all
maximal consistent subsets of Aff excluded by Ref.
Aristotle’s Early Logic
79
It is worth noting that the requirement that syllogisms be consistently premissed
bears directly on our present question. T is dignified as the proposition which Ref
refutes by virtue also of the requirement that Ref have consistent premisses. Since
Ref is a syllogism then { C, B, T} is inconsistent and { C, B} is consistent. So,
too, are {C, T} and {B, T}, since if they were not, C would entail ->T and B
would entail -i T, a happenstance precluded, each time, by Refs syllogisity. Thus
three ideas cohere: (1) the idea that T is unique in Aff precisely because it is T
that cannot be a premiss of Ref, (2) the idea that syllogisms must be consistently-
premissed; and (3) the idea that { T } is the intersection of all maximal consistent
subsets of Aff excluded by Ref.
Just how narrowly “logical” is this characterization? Or, more carefully, how
non-dialectical is it? Suppose that we said that the idea of the proposition that a
refutation refutes could be analyzed without any reference to dialectical procedure.
Then any syllogism whatever would count as the refutation of the contradictory
of its own conclusion. Every syllogism would be a refutation, and a great many
refutations would be refutations of propositions no one has ever proposed or will.
There is no great harm (in fact, there is considerable economy) in speaking this
way. But it is not Aristotle’s way. Taken his way, refutations can arise only in
question-and-answer games of the sort that we have been considering. We may
say, then, that refutations have a dialectically minimal characterization. That is,
(a) their premisses must be answers given by A to Q , and (b) what they refute
must be theses advanced by A. Nothing further is needed beyond this dialectical
minimum. In particular, there is no need to invoke the dialectical rules, Premiss-
Selection and No-Retraction. Premiss-Selection is unneeded in as much as there
is an entirely non-dialectical reason for excluding T as a premiss of any refutation
of T. As we have seen, no argument of the form
Ai
Ai
Therefore, -i/l,
is a syllogism.
Nor, as we have also seen, is the No-Retraction Rule needed to enable the
specification of T as the unique proposition refuted by a refutation. Thus the idea
of refutations as refutations of some unique T can be specified without exceeding
what we have been calling the dialectical minimum. It is in precisely this sense
that our target notion is a “broadly logical” matter.
Whether we find ourselves drifting toward a dialectical explication of that which
a refutation refutes or to a more narrowly syllogistic specification of it, it is fun¬
damentally important that on neither construal is a refutation of T definitively
probative for T. Refutations do not in the general case falsify what they refute. In
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John Woods and Andrew Irvine
contrast to this, reductio ad impossible arguments are probative, but they are not
refutations in the present sense. They lack the requisite forms. 61 It is not that
refutations fail to falsify something. They always falsify the conjunction of the
members of Aff. And it is not that refutations are indeterminate. They always
refute some unique single proposition. What refutations do not manage to do in
the general case is to bring what they falsify into alignment with what they refute.
What they falsify and what they refute are different propositions. We have said
that refutations do not falsify what they refute, in the general case. We must ask
whether there might be exceptions to this. There are. Sometimes the premisses of
a refutation, Ref 1 , are (known to be) true. If so, Ref 1 constitutes a proof of the
falsity of T, when T is the proposition that Ref 1 refutes. Does it not follow from
this that although refutations do not in general demonstrate the falsity of what
they refute, sometimes they do?
Suppose that Ref 1 is
X
Y _
Therefore, ->T
in which X and Y are (known to be) true. As before, {X, Y, T } is an inconsistent
set, but there is a difference. Ref 1 falsifies T. This makes for a curious dialectical
symmetry between A and Q. A’s obligation to answer honestly is the obligation
to offer to Q premisses which he (he., A) believes to be true. Thus for any rule-
compliant answerer against whom there is a successful refutation, Ref 1 , given what
A is required to believe, he must also believe that Ref 1 falsifies his own thesis T.
Though this is what, in all consistency, A is required to believe, it does not follow
that it is true, nor need A himself believe that its truth follows from what he is
required to believe.
Q, on the other hand, is differently positioned. Q has no role to play in the
semantic characterization of A’s answers, hence in judging the truth values of his
(he., Q’s) premisses. Of course, Q will often have his own opinions about the truth
or falsity of A’s replies, hence about the truth or falsity of his (Q’s) own premisses.
Whatever such opinions are, they have no role to play in the construction of Q’s
refutation. There is in this a strategic point. Since the success or failure of Q’s
refutation of A’s thesis, T, is independent of the truth values of the refutation’s
premisses, it might be thought that the most appropriate stance for Q to take
toward those premisses is the one that shows A to best advantage. In fact, there is
no semantic stance which Q can take towards A’s answers, because there is none
which shows A to best advantage.
61 For one thing, reductio arguments allow for a special class of non-premisses as lines in the
proof, i.e., assumptions. For another, the conclusion of a reductio is a pair {<E>, r -i4> n ), which,
though a contradiction, is not the contradictory of the proof’s assumption.
Aristotle’s Early Logic
81
11 AD HOMINEM PROOF
In several passages in On Sophistical Refutations , Aristotle seems to think that
refutations are proofs, but in a looser sense of “proof’ than reductio arguments. 62
In other places, refutations appear to be proofs in no sense of the word. 63 For
I mean, ‘proving by way of refutation’ to differ from ‘proving’ in that, in
proving, one might seem to beg the question, but where someone else is
responsible for this, there will be a refutation, not a proof (Metaphysics,
1006°, 15-18). 64
Thus
In such matters there is no proof simply, but against a particular per¬
son, there is ( Metaphysics , K5, 1062“, 2-3). 65
This is Ross. In Barnes’ version we have it that
About such matters there is no proof in the full sense, though there is
proof ad hominem. 6 **
It is hardly imaginable that there should be any contention about the origins of
the phrase “ad hominem' ’ to characterize a particular class of arguments. We owe
the concept not to Locke (as Locke himself expressly said), not to Galileo, not to
Boethius, but to Aristotle himself. It is clear that Aristotle is of two minds about
the ad hominem. He is tempted to think of ad hominem arguments both as proofs
of no kind and as proofs of some kind. Aristotle expressly contrasts arguments
“against the man” with arguments against the man’s position, and the former are
considered substandard in some way, as witness On Sophistical Refutations 20,
177 6 , 31-34, and 22, 178\ 16-23.
At times (see e.g., On Sophistical Refutations , 8, 170“, 12-19), Aristotle tries
to draw a distinction within the class of refutations between those that turn on
ad hominem moves and those that do not. The former he condemns outright as
sophistical refutations. 67
6 ~ On Sophistical Refutations 167^, 8-9 ff. Cf. Prior Analytics B27, 70°, 6-7 and Rhetoric
T13, 1414“, 31-37.
63 See On Interpretation 11, 21 tt , 5 ff. Cf. Metaphysics A5, 1015k, 8, and Posterior Analytics
A9, 76°, 13-15.
64 This is the Ross translation [Ross, 1984]. [Barnes, 1984] has it this way: “Now negative
demonstration I distinguish from demonstration proper, because in a demonstration one might
be thought to be assuming what is at issue, but if another person is responsible for the assumption
we shall have negative-proof, not demonstration.”
65 Cf. Metaphysics K5, 1062“, 30-31 and T4, 1006°, 25-26.
66 Cf. On Sophistical Refutations , 170“, 13, 17-18, 20; 177 6 , 33-34; 178 b , 8-17; 183“, 22, 24;
and Topics , $11, 161 a , 21.
67 Cf. On Sophistical Refutations , 1, 164 a , 20fF.
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Thus we have two contrasts to keep track of, and whose confusion is ruinous
for a correct understanding of Aristotle’s position. There is the contrast, 68 first,
between a proof “simply” and a refutation or argument ad hominem. In relation
to this contrast, the following things can be said: first, ad hominem arguments are
not sophistical or fallacious; and, second, ad hominem arguments are refutations,
hence not proofs “simply.” What is it to fail to be a proof simply? There are two
possibilities, and Aristotle anticipates them both. One way of not being a proof
simply is being what we have called a non-falsifying refutation. Non-falsifying
refutations are in no sense proofs against the propositions they refute. Another
way of not being a proof simply is being what we have called a falsifying refutation.
Falsifying refutations are proofs in some sense, but they are not proofs in every
sense. For example, they are not demonstrations in which there is a strict epistemic
priority rating on the premisses of the refuting syllogism.
That is the story of the first contrast. The second contrast is another matter.
It is a contrast between ad hominems in two separate senses. In the first sense,
an argument is ad hominem just in case it qualifies as a refutation. Arguments
that are ad hominem in the second sense are defective would-be ad hominems in
the first sense and, as such, reasonably can be regarded as fallacious. Aristotle
does not list the ad hominem in his catalogue of thirteen fallacies, as set out
in On Sophistical Refutations ; at least he does not give any of the thirteen the
name “ad hominem .” Even so, the account that he does give, such as it is, leaves
plenty of room to accommodate ad hominems in the second sense in the category
of ignoratio elenchi. Also possible in principle is accommodation of ad hominem
fallacies known to a much later tradition as “circumstantial,” “abusive” and “ tu
quoque .”
This does not change the fact that, for Aristotle, the dominant notion of ad
hominem argument is Lockean (if the anachornism may be forgiven). It is a
concept of ad hominem that is nicely captured by the structure of refutation. With
refutations as such there is no question of fallaciousness. The problem rather is
how closely refutations resemble proofs. Aristotle has two answers: refutations
resemble proofs not at all; and refutations resemble proofs loosely. As we see,
this is a distinction nicely preserved by the distinction between non-falsifying and
falsifying refutations.
12 SOPHISTICAL REFUTATIONS
We have already made the point that there is nothing in On Sophistical Refutations
that would qualify as a full and mature theory of any fallacy there discussed. We
have conjectured that Aristotle’s apparent theoretical indifference to the fallacies
might be an adumbration of the perfectibility thesis of Prior Analytics. For recall
that if the perfectibility thesis is true, then syllogisms turn out to be effectively
recognizable. If this is so, the distinction between genuine and only apparent
68
Metaphysics K5.
Aristotle’s Early Logic
83
syllogisms (he., fallacies) becomes wholly transparent, thus making it unnecessary
to have accounts of the thirteen fallacies in which they are effectively recognizable.
Even so, Aristotle does write at some length about the fallacies. He does so in
the context of a particular kind of argument, of which refutations are a notable
subcase. (We note in passing that fallacies are also discussed in the Analytics and
Rhetoric in the context of different kinds of arguments than those discussed in On
Sophistical Refutations, viz., demonstrations and enthymemes, respectively.) In On
Sophistical Refutations Aristotle gives fairly full accounts of sixty four 69 examples
of sophistical refutations which are only apparently syllogisms. Of these, forty-
nine have, by the lights of Prior Analytics, the wrong number of premisses, or
premisses or conclusions of the wrong sort.
Aristotle’s account of sophistical refutations begins with a discussion of “con¬
tentious arguments,” a sort of intellectual contest commonly performed in the
Greek academies, courts and councils. It is quite clear that at one level On So¬
phistical Refutations is a practical manual in which types of manoeuvres that result
in unsatisfactory resolutions of contentious arguments are identified, and methods
for spotting and blocking them are suggested; but at another level Aristotle is less
interested in the practical question of how to train people to win argumentative
contests than he is in developing a theory of objectively good reasoning.
At On Sophistical Refutations 16, 175 a , 5-17, Aristotle explains the importance
of a theory of contentious argument:
The use of [contentious arguments], then, is for philosophy, two-fold.
For in the first place, since for the most part they depend upon the
expression, they put us in a better condition for seeing in how many
ways any term is used, and what kind of resemblances and what kind
of differences occur between things and between their names. In the
second place they are useful for one’s own personal researches; for the
man who is easily committed to a fallacy by someone else, and does not
perceive it, is likely to incur this fate himself also on many occasions.
Thirdly [sic] and lastly, they further contribute to one’s reputation, viz.,
the reputation of being well trained in everything, and not experienced
in anything: for that a party to arguments should find fault with them
and yet cannot definitely point out their weakness, creates a suspicion,
making it seem as though it were not the truth of the matter but
inexperience that put him out of temper.
Aristotle is concerned to set out various ways in which a would-be refutation
fails. He classifies failed refutations into those that depend on language and those
that depend on factors external to language, although it may be closer to Aris¬
totle’s intentions here to understand the word “language” as “speech.” Aristotle
is aware that some fallacies arise because a given word may be used ambiguously.
However, when the argument in question is spoken, the offending word is given
a different pronunciation at different occurrences. Thus hearing the argument,
69 A further fifty-five examples are alluded to more briefly; see [Dorion, 1995, p. 93].
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John Woods and Andrew Irvine
rather than reading it, sometimes flags the ambiguous term and makes it easy for
the arguer to avoid the ambiguity. For such fallacies, the mediaevals used the term
in dictione, and it would appear that what is meant are fallacies whose commis¬
sion is evident by speaking the argument. Not all fallacies can be identified just
through speaking the arguments in which they occur. The mediaevals translated
Aristotle’s classification of these as extra dictionem fallacies, that is, as not being
identifiable by speaking them. On the other hand, Aristotle also says that there
are exactly six ways of producing a “false illusion in connection with language”
(165 b , 26), (emphasis added), and his list includes precisely six cases. Further,
Aristotle occasionally notices that some of his extra dictionem fallacies also qual¬
ify for consideration as language dependent, for example, ignoratio elenchi (167 a ,
35) and many questions (175 6 , 39). So the modern day practice of taking the in
dictione fallacies to be language-dependent and the extra dictionem fallacies to be
language-independent finds a certain justification in Aristotle’s text. The following
schema presents itself:
Table 3. Sophistical Refutations
In Dictione
Extra Dictionem
(1) equivocation
(7) accident
(2) amphiboly
(8) secundum quid
(3) combination of words
(9) ignoratio elenchi
(4) division of words
(10) consequent
(5) accent
(11) non-cause as cause
(6) form of expression
(12) begging the question
(13) many questions
It is immediately evident that Aristotle’s placement of these sophistries does
not fit especially well with the “discernible in speech versus not discernible in
speech” distinction. For example, equivocation involves the exploitation of a term’s
ambiguity, and can be illustrated by the following argument:
The end of life is death
Happiness is the end of life
Therefore, happiness is death.
But this is a mistake that is not necessarily made evident just by speaking the
argument. It is interesting that in this example a certain logical form is discernible,
viz.,
T is D
H is E
Therefore, H is D
Aristotle’s Early Logic
85
in which H stands for “happiness,” E for “the end of life” in the sense of the goal
or purpose of life, T for “the end of life” in the sense of the termination of life,
and D for “death.” The form is certainly invalid; it commits what later writers
would call the “fallacy of four terms.” It does not however commit the fallacy of
ambiguity, since in it the term “end” is fully disambiguated.
In contrast, amphiboly arises from what today is called syntactic (as opposed
to lexical) ambiguity, as in the sentence “Visiting relatives can be boring,” which
is ambiguous between (1) “Relatives who visit can be bores” and (2) “It can be
boring to visit relatives.” To see how amphiboly can wreck an argument, consider,
Visiting relatives can be boring
Oscar Wilde is a visiting relative
Therefore, Oscar Wilde can be boring.
If the first premiss is taken to have the meaning of (1), the argument is a syllogism.
If it is taken to have the meaning of (2), the argument is not a syllogism but a par-
alogismos, a piece of “false reasoning.” Even so, our case seems to collapse into an
ambiguity fallacy, with “visiting relatives” the offending term -ambiguous between
“the visiting of relatives” and “relatives who visit.” Here, too, an amphibolous ar¬
gument seems not to be one that an arguer would be alerted to automatically just
by speaking it, although with sufficient oral emphasis the appropriate distinctions
might be made: “VISITING relatives can be boring” may mean something quite
different than “visiting RELATIVES can be boring.”
The next two types of sophistical refutation, combination and division of words,
can be illustrated with the example of Socrates walking while sitting. Depending
on whether the words “can walk while sitting” are taken in their combined or their
divided sense, the following is true or not:
Socrates can walk while sitting.
Taken as combined, the claim is false, since it means that
Socrates has the power to walk-and-sit at the same time.
However, in their divided sense, these words express the true proposition that
Socrates, who is now sitting, has the power to stop sitting and to start
walking.
This is a better example of an in dictione fallacy. “Socrates, while sitting,
CAN WALK” sounds significantly different from “Socrates can WALK WHILE
SITTING.” In this context, it is worth noting that composition and division fal¬
lacies of the present day are not treated as fallacies in dictione [Copi and Cohen,
1990, pp. 17-20]. Rather they are understood to be fallacies that result from mis¬
managing the part-whole relationship. Thus the modern fallacy of composition is
exemplified by
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John Woods and Andrew Irvine
All the members of the Oakland As are excellent players
Therefore, the Oakland As are an excellent team.
Division fallacies make the same mistake, but in the reverse direction, so to
speak:
The As are a top-ranked team
Therefore, all the As players are top-ranked players.
We see, then, that combination and division of words is an in dictione fallacy
for Aristotle, whereas composition and division is an extra dictionem fallacy for
later writers.
Accent and form of expression, are perhaps rather difficult for the reader of
English to understand, since English is not accented in the way, for example, that
French is. It is troublesome that the Greek of Aristotle’s time was not accented
either; that is, that there were no syntactic markers of accent such as “e” (acute);
“e” (grave); and “a” (circumflex). Even so, Aristotle introduces accents in his
discussion of Homer’s poetry. Conceding that “an argument depending upon ac¬
cent is not easy to construct in unwritten discussion; in written discussion and in
poetry it is easier ” (Sophistical Refutation, 166 6 , 1-2), Aristotle notes that
some people emend Homer against those who criticize as absurd his
expression to pev ou xaTomuGeTai 6ii(3pu>. For they solve the difficulty
by a change of accent, pronouncing the 0v with an acute accent (166 6 ,
2 - 6 ).
The emendation changes the passage from “Part of which decays in the rain” to
“It does not decay in the rain,” a significant alteration to say the least.
Form of expression is meant in the sense of “shape of expression” and involves
a kind of ambiguity. Explains Aristotle,
Thus (e.g.,) ‘flourishing’ is a word which in the form of its expression is
like ‘cutting’ or ‘building’; yet the one denotes a certain quality— i.e.,
a certain condition—while the other denotes a certain action (166 & ,
16-19).
Hamblin avers with a certain pungency that “[i]t was given to J.S. Mill to make
the greatest of modern contributions to this Fallacy by perpetrating a serious
example of it himself .... He said ...
The only proof capable of being given that an object is visible, is that
people actually see it. The only proof that a sound is audible, is that
people hear it; and so of the other sources of our experience. In like
manner, I apprehend, the sole evidence it is possible to produce that
anything is desirable, is that people do actually desire it.
But to say something is visible or audible is to say that people can see or hear it,
whereas to say that something is desirable is to say that it is worthy of desire or,
Aristotle’s Early Logic
87
plainly, a good thing. Mill is misled by the termination ‘-able’ ” [Hamblin, 1970,
p. 26]. Unfortunately here, too, we seem not to have an especially convincing
example of a fallacy discernible in speech.
Turning now to extra dictionem fallacies, accident also presents the
present-day reader with a certain difficulty. The basic idea is that what can be
predicated of a given subject may not be predicable of its attributes. Aristotle
points out that although the individual named Coriscus is different from Socrates,
and although Socrates is a man, it would be an error to conclude that Coriscus is
different from a man. This hardly seems so, at least when in the conclusion “is
different from a man” is taken to mean “is not a man.” The clue to the example is
given by the name of the fallacy, “accident.” Part of what Aristotle wants to say is
that when individual X is different from individual Y, and where Y has the acci¬
dental or non-essential property P (e.g ., being six feet tall), it does not follow that
X is not six feet tall, too. But this insight is obscured by two details of Aristotle’s
example. The first is that “is a man” would seem to be an essential property of
man. Here, however, Aristotle restricts the notion of an essential property to a
synonymous property, such as “is a rational animal.” The other obscuring feature
of the case is that Aristotle also wants to emphasize that what is predicable of an
individual is not necessarily predicable of its properties. If we take “Coriscus is
different from” as predicable of Socrates, it does not follow that it is predicable of
the property man, which Socrates has. But why should this be so if no individual
(Coriscus included) is identical to any property (including the property of being
a man)? Perhaps it is possible to clarify the case by differentiating two meanings
of “is different from a man.” In the one meaning, “different from a man” is the
one-place negative predicate “is not a man”; and in its second meaning it is the
negative relational predicate “is not identical to the property of being a man.”
Thus, from the fact that Coriscus and Socrates are different men, it does not fol¬
low that Coriscus is not a man. But it does follow that Coriscus is not identical
to the property of being any man. Little of this treatment survives in present day
accounts. For example, in [Carney and Scheer, 1980, p. 72], the fallacy of accident
is just a matter of misapplying a general principle, that is, of applying it to cases
“to which they are not meant to apply.”
Secundum quid is easier to make out. “ Secundum quid’’ means “in a certain re¬
spect.” In this sense, the error that Aristotle is trying to identify involves confusing
the sense of a term in a qualified sense with its use in its absolute, or unqualified,
sense. Thus from the fact that this black man is a white-haired man, it does not
follow that he is a white man. Similarly, from the fact that something exists in
thought it does not follow that it exists in reality (Santa Claus, for example).
In its most general form, the secundum quid fallacy is the mistake of violating
the rule that, when reasoning or arguing, our claims and counterclaims about
things must honour significant similarities and must not over exploit differences.
If one party asserts “All A are B,” it is not enough for the second party to attempt
to confuse this opponent with the statement “Some A are not B," even if it is true.
Also required is that his use of the terms “A” and “£?” must agree with those of his
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John Woods and Andrew Irvine
opponents with regard to meaning, respects in which the term is applied, temporal
factors, and so on. Thus if the one party’s “All A are B” were “All bachelors are
unmarried” and his opponent’s “Some A are not B” were “Some holders of a first
university degree are married (eventually),” it would be ludicrous to suppose that,
even if true, it damages the first claim in any way.
Ignoratio elenchi, or “ignorance of what makes for a refutation,” results from
violating any of the conditions on what constitutes a proper refutation. As we
have pointed out, a refutation is genuine when one party, the questioner, is able
to fashion from the other party’s (the answerer’s) answers a syllogism whose con¬
clusion is the contradictory, not-T, of the answerer’s original thesis, T. There are
thus two ways in which the questioner might be guilty of ignoratio elenchi. He
might have made the syllogistic-mistake of supposing that not-T follows from the
premisses when it does not or, although it does follow from those premisses, one or
more of them is syllogistically impermissible. For example, Aristotle requires that
all premisses of a syllogism be propositions and, as we have pointed out, proposi¬
tions are statements in which just one thing is predicated of just one thing (On
Sophistical Refutations 169“, 8). Thus the statement, “Bob and Sally are going to
the dance,” is not a proposition, even though it clearly implies “Bob is going to
the dance,” which is a proposition. As we have seen, Aristotle has technical rea¬
sons for restricting the premisses of syllogisms to propositions; and it is clear that
he thinks that if a questioner derives a conclusion from non-propositions which
imply it, he has not constructed a syllogism. Accordingly, he has not constructed
a refutation. In fact, he has committed the fallacy of many questions (see below).
The second way in which a questioner can be guilty of ignorance of what makes
for a (genuine) refutation is when he constructs from his opponent’s answers a
faultless syllogism, but its conclusion is not the contradictory of his opponent’s
thesis, T. It thus is the mistake of supposing that a pair of propositions {P, Q} are
one another’s contradictories when they are in fact not. If the first type of error can
be called a syllogistic error , the second can be called a contradiction error [Hansen
and Pinto, 1995, p. 321], This has a bearing on how Aristotle understands the
relationship of fallacies to sophistical refutations. Some commentators hold that
fallacies and sophistical refutations are the same thing. Others are of the view that
a refutation is sophistical just because it contains a fallacy, i.e., when the would-
be syllogism that constitutes the would-be refutation commits either a syllogistic
error or a contradiction error—a logical error in each case. Aristotle even goes so
far as to suggest a precise coincidence between the in dictione — extra-dictionem
distinction and the distinction between contradiction errors and syllogistic errors:
All the types of fallacy, then, fall under ignorance of what a refutation
is, those dependent on language because the contradiction, which is the
proper mark of a refutation, is merely apparent, and the rest because of
the definition of syllogism (On Sophistical Refutations 6, 169“, 19-21;
emphasis added; cf. [Hansen and Pinto, 1995, p. 321].)
Aristotle’s Early Logic
89
In present-day treatments ( e.g ., [Copi and Cohen, 1990, pp. 105-107]), the
ignoratio elenchi is the fallacy of an argument which appears to establish a certain
conclusion, when in fact it is an argument for a different conclusion. There is
some resemblance here to Aristotle’s contradiction-error, which can be considered
a special case.
Aristotle says (On Sophistical Refutations 168“, 27; 169 & , 6) that the fallacy
of consequent is an instance of the fallacy of accident. Bearing in mind that
Aristotle thinks that consequent involves a conversion error, perhaps we can get a
clearer picture of accident. As noted above, accident is exemplified by a confusing
argument about Coriscus and Socrates. We might now represent that argument
as follows:
(1) Socrates is a man
(2) Coriscus is non-identical to Socrates
(3) Therefore, Coriscus is non-identical to a man.
In line (1), the word “is” occurs as the is-of-predication. Suppose that line (1)
were in fact convertible , that is, that (1) itself implied
(1') A man is Socrates.
In that case, the “is” of (1) would be the is-of-identity, not the is-of-predication,
and the argument in question would have the valid form
(T) S = M
(2') C ± S
(3') Therefore, C ?M.
Thus the idea that (1) is convertible and the idea that its “is” is the is-of-
identity come to the same thing and this is the source of the error. For the only
interpretation under which
Socrates is a man
is true, is when “is” is taken non-convertibly, i.e., not as the is-of-identity, but as
the is-of-predication.
Consequent is an early version of what has come to be known as the fallacy
of affirming the consequent [Copi and Cohen, 1990, pp. 211, 282]. In present-
day treatments this is the mistake of concluding that P on the basis of the two
premisses, “If P then Q,” and Q. Where P is the antecedent and Q the consequent
of the first premiss, the fallacy is that of accepting P on the basis of having affirmed
the consequent Q. Aristotle seems to have this kind of case firmly in mind; but he
also thinks of consequent as a conversion fallacy, that is, as the mistake of inferring
“All P are S" (“All mortal things are men”) from “All S are P” (“All men are
mortal”).
Non-cause as Cause also appears to have been given two different analyses.
In the Rhetoric it is the error that later writers call the fallacy of post hoc, ergo
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John Woods and Andrew Irvine
propter hoc, the error of inferring that event e is the efficient cause of event e' just
because the occurrence of e' followed upon (temporally speaking) the occurrence
of e. In On Sophistical Refutations, however, it is clear that Aristotle means by
“cause” something like “reason for.” In this case, the non-cause as cause error is
exemplified by the following type of case. Suppose that
P
_Q _
Therefore, -T
is a refutation of the thesis T. Then the argument in question is a syllogism, hence
a valid argument. As any reader of modern logic knows, if the argument at hand
is valid, then so too is the second argument,
R
P
_Q _
Therefore, ->T
no matter what premiss R expresses. But this second argument is not a syllogism
since a proper subset of its premisses, namely, {P, Q}, also entails its conclusion.
Hence our second argument cannot be a refutation. This matters in the following
way: Aristotle thinks of the premisses of a refutation as reasons for (“causes of’)
its conclusion; but since our second argument is not a refutation of T, R cannot
be a reason for not-T.
In the course of real-life contentions, an answerer will often supply the questioner
with many more answers than the questioner can use as premisses of his would-be
refutation. Aristotle requires that syllogisms have no idle premisses. Thus the
questioner is obliged to select from the set of his opponents’ answers just those
propositions, no more and no fewer, than non-circularly necessitate the required
conclusion.
In this sense, the fallacy of non-cause as cause is clearly the mistake of using an
idle premiss, but it may not be clear as to why Aristotle speaks of this as the error
in which a non-cause masquerades as a cause. Something of Aristotle’s intention
may be inferred from a passage in the Physics (195°, 15), in which it is suggested
that in syllogisms premisses are the material causes (the stuff) of their conclusions,
i.e., that premisses stand to conclusions as parts to wholes, and hence are causes
of the whole. Idle premisses fail to qualify as material causes; they can be removed
from an argument without damaging the residual sub-argument. Real premisses
are different. Take any syllogism and remove from it any (real) premisses and the
whole (i.e., the syllogism itself) is destroyed. In other places, Aristotle suggests
a less technical interpretation of the fallacy, in which the trouble with R would
simply be its falsity, and the trouble with the argument accordingly would be the
derivation of not-T from a falsehood—a false cause (On Sophistical Refutations
167 b , 21). 70
70 Non-cause is discussed in greater detail and is given a somewhat different emphasis in [Woods
and Hansen, 2003].
Aristotle’s Early Logic
91
Aristotle provides several different treatments of begging the question , or petitio
principii. In On Sophistical Refutations it is a flat-out violation of the definition
of “syllogism” (hence of “refutation”). If what is to be proved is also assumed as
a premiss, then that premiss is repeated as the conclusion, and the argument in
question fails to be a syllogism. Hence it cannot be a refutation. On the other
hand, in the Posterior Analytics 86 a , 21, begging the question is a demonstration
error. Demonstrations are deductions from first principles. First principles are
themselves indemonstrable, and in any demonstration every succeeding step is less
certain than preceding steps; but if one inserts the proposition to be demonstrated
among the premisses, it cannot be the case that all premisses are more certain than
the conclusion. Hence the argument in question is a failed demonstration.
Aristotle recognizes five ways in which a question can be begged ( Topics 162 6 ,
34-163“, 2):
People appear to beg their original question in five ways: the first
and most obvious being if anyone begs the actual point requiring to
be proved: this is easily detected when put in so many words; but it
is more apt to escape detection in the case of synonyms, and where a
name and an account mean the same thing. A second way occurs when¬
ever anyone begs universally something which he has to demonstrate
in a particular case ...
The third way is to beg a particular case of what should be shown universally;
the fourth is begging “a conjunctive conclusion piecemeal” [Hamblin, 1970, p. 74];
and the fifth is begging a proposition from a proposition equivalent to it.
Contemporary readers are likely to find the expression “begging the question”
rather obscure (and oddly dramatic). There is no beggary to begging the ques¬
tion, other than the questioner’s soliciting a premiss for his evolving syllogism by
putting to his opponent a Yes-No question. Further, Aristotle actually collapses
the intuitive distinction between the question that produces the answer and the
premiss that that answer is eligible to be. That is, what Aristotle calls a question
is in this context the premiss produced by the answer to it. Strictly speaking,
then, begging such a question is using it as a premiss in a would-be syllogism. The
“original question” of the two parties is the answerer’s thesis, T. An opponent
would beg the question in the first way if he begged the point required to be shown,
i.e., not-T. But if not-T were indeed a premiss of any deduction whose conclusion
were also not-T, the deduction would be circular, as we have said; hence it would
be neither a syllogism nor a refutation. Aristotle rightly notices that such cases
are unlikely to fool actual reasoners, but he reminds us that if a synonym of not-T
were used as a premiss, then the premiss would look different from the conclu¬
sion, and the circularity might go undetected. In fact, this seems also to be what
Aristotle has against the fifth way, i.e., deducing a proposition from one equivalent
to it.
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John Woods and Andrew Irvine
In the second way of begging the question, Aristotle has in mind a certain
form of what came to be called “immediate inference.” It is exemplified by the
subalternation argument
Since all A are B , some A are B.
Aside from the fact that single-premiss arguments seem not to qualify as syl¬
logisms, it is difficult to make out a logical fault here. Bearing in mind that
the fault is the questioner’s (i.e., premiss selector’s) fault, not the answerer’s,
one might wonder what is wrong with a questioner’s asking a question which, if
answered affirmatively, would give him a desired conclusion in just one step. Ev¬
idently Aristotle thinks that a refutation is worth having only if every premiss
(individually) is consistent with the answerer’s thesis. Such is Ross’ view of the
matter:
And syllogism is distinguished from petitio principii in this, that while
in the former both premises together imply the conclusion in the latter
one premise alone does so [Ross, 1953, p. 38].
Thus in a good refutation, the thesis is refuted, never mind that the thesis is
consistent with each separate answer given. (See [Woods, 2001, ch. 9].)
The third way of begging seems quite straightforward. It is illustrated by the
plainly invalid form of argument
Since some A are B, all A are B.
Even so, as the example makes clear, begging the question is fundamentally an
error of premiss selection, given that the answerer’s job is to elicit premisses that
syllogistically imply its contradictory, “All A are BA Again, begging the question
is selecting a premiss, and in the present case, the questioner has begged the wrong
question, i.e., he has selected the wrong premiss. It is a premiss which does the
answerer’s thesis no damage; and in any extension of this continuing argument in
which damage were to be done, this premiss “Some A are B" would prove idle.
Genuinely perplexing is the fourth way of begging the question. By the require¬
ment that syllogisms be constructed from propositions, it would appear that no
competent syllogizer would ever wish to conclude his argument with a conjunctive
statement (since these are not propositions). However, consider the argument
P
_Q _
Therefore, P and Q.
It is clearly valid. What Aristotle appears to have in mind is that it is useless for
the questioner first to beg for P, then for Q, if his intention is to conclude “P and
Q.” The manifest validity of the argument might deceive someone into thinking
he had produced a syllogism, but the fault lies less with his premiss selection than
Aristotle’s Early Logic
93
with his choice of target conclusion. Once begged, those questions will assuredly
“get” him that conclusion, but it is a statement of a type that guarantees that his
argument nevertheless is not a syllogism.
The problem of would-be refutations that derive their targets in one step from
a single premiss is, according to Aristotle, the problem that the argument
All A are B
Therefore, some A are B
begs the question. The contradictory of its conclusion is inconsistent with each
of the premisses (of which there happens to be only one). If it is correct to say
that the contradictory of a syllogism’s conclusion must be consistent with each
premiss, then our argument is not a syllogism. On the other hand, Aristotle in
places appears to accept subalternative arguments ( e.g ., at Topics 119°, 32-36).
Some writers interpret Aristotle in a different way, as claiming an epistemic fallacy:
one could not know the premiss to be true without knowing the conclusion to be
true.
This raises a further matter as to whether, e.g.,
All A are B
All C are A
Therefore, all C are B
does not also beg the question. Mill is often said to have held that this is precisely
the case with all syllogisms. This was not, in fact, Mill’s view, but Aristotle seems
to consider (and reject) the possibility ( Posterior Analytics 72 b , 5-73“, 20). If the
first interpretation is correct, then were syllogisms fallaciously question-begging
as such, it could not be for the reason that affects subalternation arguments. For
All A are B
All C are A _
Therefore, all C are B
cannot be a syllogism unless its premisses fail individually to derive its conclusion;
and it cannot commit a petitio of the subalternation variety unless one of its
premisses does indeed yield the conclusion on its own.
As was said in the above discussion of ignoratio elenchi, Aristotle’s fallacy of
many questions is a very different thing from that presented in present day logic
textbooks e.g., [Copi and Cohen, 1990, pp. 96-97]. In such treatments, the fallacy
is typified by such questions as, “Have you stopped beating your dog?,” in which
there is a unconceded presupposition, namely, “The addressee has been a beater
of his dog in the past.” Aristotle intended something quite different by the many
questions fallacy. It is the error of admitting to the premiss set of a would-be
syllogism a statement that is not a proposition, in Aristotle’s technical sense of
“one thing predicated of one thing.” It was mentioned above that Aristotle had
technical reasons for requiring syllogisms to be made up of propositions. This can
94
John Woods and Andrew Irvine
be explained as follows. In the Topics (100 a , 18-21) and On Sophistical Refutations
(183°, 37-36), Aristotle declares that his aim is to discover a method (or faculty of
reasoning) from which we will be able to reason [syllogistically] about every issue
from endoxa, i.e., reputable premisses, and when compelled to defend a position,
we say nothing to contradict ourselves.
In other places, his aims are forwarded more ambitiously. At On
Sophistical Refutations 170“, 38 and 171 b , 6-7, Aristotle says that the strate¬
gies he has worked out will enable a person to reason correctly about anything
whatever , independently of knowledge of its subject matter. This is precisely what
the Sophists also claimed to be able to do. Aristotle scorns their claim, not be¬
cause it is unrealizable, but because the Sophists lack the theoretical wherewithal
to bring it off. The requisite theoretical wherewithal Aristotle took to be the
logic of syllogisms. In various respects Aristotle’s boast seems incredible. For one
thing, are there not far too many arguments, some of considerable complexity,
for any one theory to capture in their totality? Part of Aristotle’s answer lies in
a claim advanced in On Interpretation ( e.g ., 16 a , 19-26; 16 6 , 6-10, 19-25; 16 6 ,
26-17°, 2). There he asserts that anything stateable in any natural language such
as Greek, can be expressed without relevant loss in a proper sublanguage made
up exclusively of propositions, i.e., statements that say one thing of one thing.
Propositions may be said to be statements whose only logical particles are at most
the quantifier expressions “all,” “some,” “no,” “some—not,” and the connective
“not.” In particular, compound statements held together by connectives such as
“and,” “or” and “if... then,” fail to qualify. Aristotle in effect is confining the class
of propositions to the class of categorical statements: “All A are B," “Some A are
£?,” “Some A are not B," and “No A are B.” Suppose that a theorist wanted to
produce a complete grammar for all the declarative sentences of Greek. If Aristo¬
tle’s claim in On Interpretation is true, the theorist would succeed in his task if he
could produce a complete theory of these four categorical forms. This very striking
economy is also passed on to the claim that a theory of all good deductive reason¬
ing is possible. By deductive reasoning, Aristotle means reasoning expressible in
syllogisms, and syllogisms are made up of just three propositions, two premisses
and a conclusion. Further, in every syllogism, there occurs exactly one more term
than there are premisses. Thus any would-be syllogism will be made up of just
two premisses and just three terms, say, “A,” “B,” and “C.” There are just four
propositional forms, and for each of the three terms only a low finite number of
distributions of them in triples of those forms. Hence a complete theory of good
reasoning ( syllogismos ) is possible because its domain is, as we now see, low finite,
i.e., it permits exhaustive examination.
In present-day treatments, the many questions fallacy is committed by asking a
certain type of question, e.g., “Have you stopped beating your dog?” In Aristotle’s
view, the fallacy is that of using an answer to the question as a premiss. Suppose
that answer is “No.” This is equivalent to
It is not the case that (I have beaten my dog in the past and I do so
at present)
Aristotle’s Early Logic
95
or
Either I have not beaten my dog in the past or I do not do so at present.
In each case, the answer contains a connective other than “not”—“and” in the first
instance, and “or” in the second. In neither case, then is the answer a proposition
in which “one thing is said of one thing,” so it is inadmissible as a premiss of a
syllogism. Where the modern theorist sees the fallacy as an interrogative fallacy
[Hintikka, 1987], for Aristotle it is a syllogistic error.
Although On Sophistical Refutations is the primary source of what people have
come to call Aristotle’s fallacies, Aristotle gives them a somewhat different char¬
acterization in his other writings. In the Topics , we read that an argument (not
necessarily a refutation) is fallacious in four different ways: (1) when it appears
valid but is invalid in fact; (2) when it is valid but reaches “the wrong conclusion”;
(3) when it is valid but the conclusion is derived from “inappropriate” premisses;
and (4) when, although valid, the conclusion is reached from false premisses. Case
(1) might well be exemplified by the fallacy of affirming the consequent. Case (2)
might be thought of in this way: let the premisses all be drawn from the discipline
of economics, and let the conclusion be the logical truth, “Either it will rain today
or it will not.” Although that conclusion does follow validly from those premisses—
at least by modern lights—it might be objected that it is the “wrong thing” to
conclude from those premisses. Case (3) is similar. Aristotle’s own example is
one in which it is concluded that walking after a meal is not good for one’s health
(a conclusion from the art or discipline of medicine) from the premiss, proposed
by Zeno, that motion (hence walking after a meal) is impossible. Even if Zeno’s
paradoxical proposition were true, Aristotle would claim it to be an inappropriate
premiss for a medical argument, since it is not a medical premiss. Case (4) is ob¬
vious: though true conclusions are often compatible with false premisses, no true
conclusion can be established by false premisses.
The following is a list, taken from [Hansen and Pinto, 1995, p. 9] of where in
Aristotle’s writings the individual fallacies are discussed:
• Equivocation: Soph. Ref. , 4 (165\ 31 166 a , 7); 6 (168 a , 24); 7 (169 a , 22-25);
17 (175°, 36-175 6 , 8); 19; 23 (179 a , 15-19); Rhetoric II, 24 (1401 a , 13-23).
• Amphiboly: Soph. Ref, 4 (166 a , 7-22); 7 (169°, 22-25); 17 (175 Q , 36-175 6 ,
19); 23 (179 a , 19-25).
• Combination of words: Soph. Ref., 4 (166 a , 23-32); 6 (168 n , 22-25); 7 (169 Q ,
25-27); 20; 23 (179 a , 12-13); Rhet. II24 (1401 a , 24-1401 6 , 3).
• Division of words: Soph. Ref, 4 (166°, 33-39); 6 (166°, 27); 7 (169“, 25-27);
20; 23 (179“, 12-13); Rhet. II, 24 (1401°, 24-140l\ 3).
• Accent: Soph. Ref., 4 (166 6 , 1-9); 6 (168°, 27); 7 (169 a , 27-29); 21; 23
(179 a , 13-14).
96
John Woods and Andrew Irvine
• Forms of expression: Soph. Ref., 4 (166\ 10-19); 6 (168 a , 25); 7 (169°,
30-169 6 , 3), 22; 23 (179°, 20-25).
• Accident: Soph. Ref., 5 (166\ 28-37); 6 (168°, 34-168 6 , 10; 168*’, 26-169 a ,
5); 7 (169\ 3-6); 24; Rhet. II, 24 (1401 6 , 5-19).
• Secundum Quid: Soph. Ref., 5 (166*, 38-167°, 20); 6 (168 6 , 11-16); 7 (169 b ,
9-13); 25; Rhet. II, 24 (140l\ 35-1402°, 28).
• Ignoratio Elenchi: Soph. Ref. 5 (167°, 21-36); 6 (168 6 , 17-21); 7 (169 6 ,
9-13); 26.
• Consequent: Soph. Ref, 5 (167\ 1-20); 6 (168\ 26-169°, 5); 7 (169 b , 3-9);
Pr. Anal. B, 16 (64 6 , 33); Rhet. 1124 (140l\ 10-14, 20-29).
• Non-cause: Soph. Ref, 5 (167 6 , 21-37); 6 (168 6 , 22-26); 7 (169 6 , 13-17); 29;
Pr. Anal. B II 17; Rhet. II24, (1401 6 , 30-34).
• Begging the Question: Soph. Ref., 5 (167°, 37-40); 6 (168 6 , 25-27); 7 (169 6 ,
13-17); 17 (176°, 27-32); 27; Topics, 8 (161 b , 11-18); (162\ 34-163°, 13);
13 (162 fc , 34-163°, 28); Pr. Anal., 24 (41 b , 9); Pr. Anal. B, 16 (64 6 , 28-65°,
37).
• Many Questions: Soph. Ref, 5 (167 6 , 38-168°, 17); 6 (169°, 6-18); 7 (169 fc ,
13-17); 17 (175 6 , 39-176°, 19); 30.
In bringing this chapter to a close, we revisit Hamblin’s harsh remarks on what
he calls the Standard Treatment of the fallacies in which
a writer throws away all logic and keeps the reader’s attention, if at all,
only by retailing the traditional puns, anecdotes, and witless examples
of his forebears. ‘Everything that runs has feet; the river runs; there¬
fore, the river has feet’—this is a medieval example, but the modern
ones are no better [Hamblin, 1970, p. 12].
Such treatments, says Hamblin, are useless and they leave us in a situation in
which “[w]e have no theory of fallacies at all ....” [Hamblin, 1970, p. 11]. Whatever
one thinks of these complaints, it seems that Hamblin would have been entirely
happy to put Aristotle at the very top of the list of those whose views disappoint
him so. In fact, this is not what Hamblin does. Instead he excoriates many a
later writer for failing to pay due attention to Aristotle. In fairness, Hamblin
does not think uniformly well of Aristotle’s analyses. Certainly there is in these
writings no wholly developed theory of fallacies, and some of the examples are
rather silly. Even so, Hamblin thinks that Aristotle was on to something genuinely
important in two respects. First, Aristotle was, in On Sophistical Refutations,
attempting to work his way through the old subject of dialectics to the genuinely
new discipline of logic. But, secondly and rather strangely, Hamblin commends
Aristotle’s Early Logic
97
us to Aristotle’s example in treating the fallacies as inherently dialectical entities
(although Hamblin also thinks that on this point Aristotle showed signs of wavering
[Hamblin, 1970, pp. 65-66]). It is worth noting that Aristotle has an answer to
the objection that his examples of the fallacies are silly, and that they would fool
no one. For if a fallacy is an inapparently bad argument, then it is hard to see
how one could give convincing examples of such things. Presumably a good or
non-silly example would have to be one which the reader would not recognize as
a fallacy, and so in one clear sense of the term would also be a bad example.
ACKNOWLEDGEMENTS
Parts of this chapter draw upon or are adapted from John Woods’ monograph
Aristotle’s Earlier Logic, published by Hermes Science in 2001, and his entry
“Aristotle” in the File of Fallacies section of Argumentation, 13 (1999), 317-334.
We are grateful for permission to republish. We also wish to thank for their sup¬
port and sage advice, Julius Moravcsik, Hans Hansen, Lawrence Powers, David
Hitchcock, Erik Krabbe, George Boger and Dov Gabbay.
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ARISTOTLE’S UNDERLYING LOGIC
George Boger
1 INTRODUCTION
1.1 Aristotle’s commitment to reason and scientific understanding
The enlightenment thinkers of the ancient fifth century were natural heirs to the
earlier thinkers who had aimed to replace mythopoeic and religious explanation
of material phenomena with extensive observation and naturalistic explanation.
With the advent of democracy and an increasing market economy there came a
new spirit of inquiry, a new reliance on reasoned argumentation, and a new com¬
mitment to understanding nature. The aspirations of the enlightenment activists
blossomed in the humanist ideals of scientific rationalism and political liberty, in
profound philosophical inquiry into metaphysics, epistemology and ethics, and in
artistic and cultural works of enduring value. These ancients condemned super¬
stition in a way that recalls Hume’s exhortation to employ philosophy as “the
sovereign antidote ... against that pestilent distemper ... to restore men to
their native liberty”. We might recall Heraclitus’s own exhortation to listen to
the logos, the aims of the Pythagorean mathematikoi, the nous of Anaxagoras,
and the condemnation of superstition by Hippocrates in The Sacred Disease and
Ancient Medicine. Sophocles especially captured this spirit in Antigone (332-375)
where he has the Chorus sing that “there are many wonders, but nothing is more
wonderful than a human being”. Indeed, Prometheus might well have been their
patron saint, because, in providing humans with various technai, he affirmed opti¬
mism about their future. A principle underlying this optimism holds that human
beings can understand themselves and nature sufficiently to govern their own des¬
tinies without the external and apparently capricious interventions of supernatural
beings. These ancients embraced Kant’s dictum, expressed many centuries later,
“Sapere aude!" — 11 Dare to know!”. Their steady strides in the second half of the
fifth century toward consolidating and rationally organizing the sciences helped to
bring the earlier inquiries to fruition and prepared the way for the enduring ac¬
complishments of the later philosophers, scientists, and political theorists. Many
high points mark the achievements of this enlightenment, but two stand out for
their progressive humanist ambition. As Protagoras in relation to the social world
developed a political techne that affirmed the teachability of virtue and citizenship
and thus promoted an empowering democratic activism, so Hippocrates in relation
to the natural world developed a medical techne that affirmed the intelligence of
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
© 2004 Elsevier BV. All rights reserved.
102
George Boger
human beings to intervene in the workings of nature to preserve health and to
prevent disease.
Aristotle, interestingly himself the son of a physician, is an exemplary fourth
century heir to enlightenment trends in science and philosophy. He affirmed the
principle that nature in its diversity and human beings in their complexity are
comprehensible. In Metaphysics 1.2 he openly avows a humanist ideal kindred to
that of Hippocrates in Ancient Medicine. 1
The acquisition of this knowledge [sjuaxrjpr) ( episteme )] ... [has been]
regarded as not suited for man. ... God alone may have this prerog¬
ative, and it is fitting that a man should seek only such knowledge as
becomes him [and not, as the poets say, arouse the gods’jealousy]. But
we should not believe in divine jealousy; for it is proverbial that bards
tell many lies, and we ought to regard nothing more worthy of honor
than such knowledge. 2 (982b28-983a7)
Aristotle in the fourth century embraced the earlier enlightenment’s daring to know
and its optimistic confidence in reason’s ability to establish objective knowledge.
We can thus appreciate his exhortation in Nicomachean Ethics 10.7 that “we not
follow the proverb-writers to ‘think mortal thoughts’ ... Rather, as far as we
can, we ought to strive to be immortal and to go to all lengths to live a life that
expresses our supreme element” (1177b31-34). This supreme element consists
precisely in the capacity of intellect by which human beings make both themselves
and nature objects of contemplation.
Aristotle boldly began Metaphysics by affirming that “all men naturally desire
to know”. He then traced the acquisition of knowledge from sensation through
memory of the same thing and finally to art and science {episteme), which are
produced through extensive experience.
Art [tex vt )] i s born when out of many bits of information derived from
experience there emerges a grasp of those similarities in view of which
they are a unified whole. Thus, a man is experienced who knows that
'Consider a typical passage from Ancient Medicine (AM) on the scientific spirit and from
The Sacred Disease ( SD ) against superstition. “Some physicians and scientists say that it would
be impossible for anyone to know medicine who does not know what species-man consists of,
this knowledge being essential for giving patients correct medical treatment. The question that
they raise, however, is ... [wholly an abstract] matter [fit only] for [the likes of] ‘philosophy’ ... I
consider, first, that all that has been said or written by scientist or physician about natural science
has less to do with medicine than it has with the art of writing [or painting]. Next, I consider
that clear knowledge of nature can be derived from no other source except from medicine” (AM
20). “[The so-called sacred disease] is not any more divine or more sacred than other diseases,
but has a natural cause, and its supposed divine origin is due to men’s inexperience, and to
their wonder at its peculiar character. .. . Men continue to believe in its divine origin because
they are at a loss to understand it ... But if it is to be considered divine just because it is
wonderful, there will be not one sacred disease but many .. . other diseases are no less wonderful
and portentous” (SD 1, 2-14). Similar passages in The Art express this same spirit.
2 We use R. Hope’s 1960 translation of Metaphysics with modification. All emphases in
Aristotle’s texts cited here and below are added unless otherwise indicated.
Aristotle’s Underlying Logic
103
when Callias was ill of this disease he was helped by this medicine, and
so for Socrates and for many others, one by one; but to have art is to
grasp that all members of the group of those who are ill of this disease
have been helped by this medicine.
Now experience [epmsiplot] seems in no respect inferior to art in a situ¬
ation in which something is to be done. ... The reason is that experi¬
ence, like action or production, deals with things severally as concrete
individuals, whereas art deals with them generally. Thus, a physician
does not cure species-man (except incidentally), but he cures Callias,
Socrates, or some other individual with a proper name, each of whom
happens to be a man. If, then, someone lacking experience, but know¬
ing the general principles of the art, sizes up a situation as a whole, he
will often, because he is ignorant of the individuals within that whole,
miss the mark and fail to cure; for it is the individual that must be
cured.
Nevertheless, we believe that knowing and understanding [to yc eiSevca
xca to Eitaleiv] characterize art rather than experience. And so we
take experts [roue; TeyvlTac;] in an art to be wiser than men of mere
experience; because wisdom presumably comes only with knowledge,
and we believe that the experts can analyze and explain, whereas others
cannot. Men of experience discern the fact “that”, but not the reason
“why”. Hence we also hold master workmen [xouc; apxtcEXTOvac;] in
each craft to be more valuable and discerning and wise than manual
laborers [xwv xs<-P 0T£ X v “ v ]> because the former can discriminate the
various factors relevant to the various effects produced; whereas the
latter, like inanimate objects, produce effects, as fire burns, without
knowing what they are doing. Inanimate objects produce their effects
somehow by nature; and manual workers, by habit. Master workers
are presumably wiser, then, not because they are practical, but because
they have their reasons and can explain what they are doing [aXXa xorca
to Xoyov exeiv auxouc; xoci xa<; aixlat; yvopiCeiv]. (981a5-981b6)
Notwithstanding a class supremacy expressed here, it is evident that Aristotle
was animated by a firm commitment to the centrality of reason in human life.
Indeed, his many treatises on natural science, metaphysics, ethics and politics give
expression to his commanding commitments to discovering truth and establishing
knowledge. Thus, in spite of his frequent complaints about Socrates, Aristotle
nevertheless embraced his teaching in Phaedo (89d) that “there is no worse sin
than misology” and in Apology (38a) that “the unexamined life is not worth living
for a human being”. The lessons of Nicomachean Ethics require a life of reason
for realizing one’s humanity and achieving happiness. Human virtue consists in
making excellent the soul’s deliberative and scientific faculties: practical wisdom
(cppovrjmc;) “is a state grasping the truth, involving reason [piExa Xoyou], concerned
with action about what is good and bad for a human being” (NE 6.5: 1140b4-6);
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and wisdom (oocpia) “is understanding plus scientific knowledge [vouc; xai eiucmrjpr)]
of the most honorable things” (NE 6.7: 1141al8-20; cf. Meta. 982a4-6). Human
happiness requires a philosophic life. When we consider Aristotle’s bold statements
in On the Soul 3.5 and 3.7 that “actual knowledge is identical with its object [to S’
auxo eaxiv f) xax’ evepysiav ETuaxrjpr) xw Ttpaypaxi]” (430al9-20 & 431al-2), we can
more fully appreciate his exhortation in Nicomachean Ethics 10.7 that reaffirms
the spirit of his inquiry in Metaphysics 1.2 not to bow to ignorance and inability,
but always “to live a life that expresses our supreme element”. Prometheus may
have stolen for us fire from the hearth of the Olympians, but Aristotle aimed to
secure for us a place at their table. In this connection, then, Aristotle’s logical
investigations are among his enduring accomplishments toward realizing this end.
1.2 Previous interpretations
Until recently the difference between traditional or ‘Aristotelian’ logic and Aristo¬
tle’s own ancient logic had been blurred. This is similar to the blurring of a similar
distinction between Christian religion and the teachings of Jesus, or the difference
between various ‘Marxian’ philosophies and the teachings of Karl Marx. It is re¬
markable, for example, that for Aristotle in every syllogism the conclusion follows
logically from the premisses. This contrasts with the usage of traditional logicians,
who continue to speak of invalid syllogisms. For Aristotle this is a contradiction in
terms, an oxymoron. In addition, Aristotle would never have tested the validity or
invalidity of a syllogism according to rules of quality, quantity, and distribution.
He had his own methods for establishing validity and invalidity. However, it was
really not possible meaningfully to distinguish the historical logic of Aristotle from
its later accretions and compare the two until modern logicians examined Aris¬
totle’s syllogistic through the lens of mathematical logic — that is, until modern
logicians turned their attention specifically to the formal aspects of deductive dis¬
courses apart from their subject matters. As a result, studies of Aristotle’s logic
since the early 20 th century have established his genius as a logician of considerable
originality and insight. Indeed, we can now recognize many aspects of his logical
investigations that are themselves modern, in the sense that modern logicians are
making discoveries that Aristotle had already made or had anticipated. Perhaps
the longevity of this oversight about the nature and accomplishments of his logi¬
cal investigations is attributable to scholars not having recognized that Aristotle
expressly treated the deduction process itself.
Jan Lukasiewicz initiated the reassessment of Aristotle’s syllogistic in the 1920s.
He was followed by James W. Miller, I. M. Bochenski, and Gunther Patzig among
others. This reassessment culminated in the 1970s and 1980s with the works of
John Corcoran, Timothy Smiley, and Robin Smith. These modern logicians used
mathematical logic to model Aristotle’s logic and discovered a logical sophisti¬
cation long overlooked by traditionalist logicians such as R. Whately, H. W. B.
Joseph, J. N. Keynes, W. D. Ross, and R. M. Eaton. These traditionalists, whose
modern origin can be traced to the Port Royal Logic , believe that Aristotle com-
Aristotle’s Underlying Logic
105
posed Prior Analytics as a logic manual for studying categorical arguments or
syllogisms. They take a syllogism to be a fully interpreted premiss-conclusion
argument whose validity or invalidity is determined by applying rules of quality,
quantity, and distribution, all of which really only help to define a syllogism. How¬
ever, traditionalists tend to conflate this sense of a syllogism with another sense
when they take a syllogism also to be a relatively uninterpreted argument pattern
whose instances are valid or invalid arguments.
Now, in spite of their equally criticizing traditionalist interpreters, mathemati¬
cal logicians themselves tend to fall into two camps concerning Aristotle’s project
in Prior Analytics. In fact, when modern logicians mathematically modeled Aris¬
totle’s logic, they tacitly distinguished two tendencies in the traditionalist inter¬
pretation, the one treating what it believed were Aristotle’s axiomatic interests,
the other treating Aristotle’s argumental interests. The axiomaticist interpreta¬
tion by Lukasiewicz, Bochenski, Miller, and Patzig takes a syllogism to be a single,
logically true conditional proposition, some of which are taken to be axioms. On
this interpretation Prior Analytics contains an axiomatized deductive system with
an implicit underlying propositional logic. Euclid’s Elements is an ancient ana¬
logue. The axiomaticists examine Aristotle’s syllogistic mathematically from a
Frege-Russell view of logic as formal ontology. On the other hand, deductionists
examine Aristotle’s logic mathematically from a Quinian view of logic as formal
epistemology. 3 The deductionist interpretation of Corcoran, Smiley, and Smith
takes a syllogism to be a deduction , that is, to be a fully interpreted argumenta¬
tion having a cogent chain of reasoning in addition to premisses and a conclusion.
On this interpretation the number of premisses is not restricted to two. This in¬
terpretation sees Prior Analytics as having proof-theoretic interests relating to a
natural deduction system. Interpretive lines, then, are drawn along what each view
considers a syllogism to be and what each takes to be Aristotle’s accomplishment
in Prior Analytics.
However, notwithstanding significant differences among modern interpretations,
there are two striking similarities. (1) All three interpretations consider the pro¬
cess of reduction (avaycoyf); avdyeiv) treated in Prior Analytics A 7 in virtually
the same way. The various interpreters hold that reduction amounts to deduction
of some syllogisms, taken as derived, from others, taken as primitive, to form a
deductive system. In addition, they do not distinguish reduction from analysis
(dvdXuCTic;; dvaXuetv). Aristotle, though, distinguished deduction from reduction
and each of these from analysis. (2) The axiomaticists and deductionists equally
consider Aristotle to have employed the method of counterargument to establish
knowledge of invalidity in his treatment of syllogisms in Prior Analytics AJf-6.
However, Aristotle there used neither the method of counterargument nor the
3 J. Corcoran (1994) clarifies two approaches to logic that have been the vantage points of
modern interpretations using the theoretical apparatus of mathematical logic. From the view¬
point of formal ontology, logic investigates certain general aspects of reality, and so Aristotle is
seen to deduce laws of logic from axiomatic origins; he is concerned with logical truths. From the
viewpoint of formal epistemology, logic amounts to an investigation of deductive reasoning
per se, and so Aristotle is seen to describe deductions and the process of deduction.
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George Boger
method of counterinterpretation. It is astonishing that such different interpreta¬
tions of a syllogism could produce such similar views about the logical relationships
among the syllogisms.
In great measure, interpretive problems are attributable to scholars not having
sufficiently recognized Aristotle’s acumen in distinguishing logical and metalogical
discourses. Deductions are equally performed in different languages: (1) in an
object language about a given subject matter and (2) in a metalanguage, which is
used to model formal aspects of object language discourses relating to sentences,
arguments, argumentations and deductions. Discourses in these categorially dif¬
ferent languages may or may not use the same deduction system or the same logic
terms with the same or different denotations. We distinguish an object language
deduction from a metalogical deduction. Aristotle understood his syllogistic de¬
duction system to function at both levels. In Prior Analytics he both studied his
syllogistic logic and its applications and he used this logic in that study. Tradi¬
tionalists, however, altogether missed Aristotle’s making this distinction by their
conflating two senses of a syllogism and, consequently, they overlooked a syllogis¬
tic deduction process. Axiomaticists mistook a conditional sentence corresponding
to a syllogism for the syllogism itself to confuse the two levels of discourse and
thereby they lost sight of Aristotle’s principal concern with deduction. Still, they
were correct to focus on his metalogical treatment of ‘syllogistic forms’, even if in
their enthusiasm to apply mathematical logic to Aristotle’s work they mistakenly
saw an axiomatized deductive system in Prior Analytics. Deductionists correctly
focused attention on Aristotle’s concern with the process of deduction and a natu¬
ral deduction system. However, in reacting to the axiomaticists, they did not take
Aristotle as himself modeling object language discourses by means of a metalogical
discourse. Nor, then, did they consider his metalogical discourse to be sufficiently
formal for his having distinguished logical syntax from semantics. Deductionists
modeled Aristotle’s logic but did not recognize Aristotle as himself providing an
ancient model of an underlying logic with a formal language.
1.3 Aristotle’s project: to establish an underlying logic
Aristotle would have agreed with Alonzo Church that “(formal) logic is concerned
with the analysis of sentences or of propositions and of proof with attention to
the form in abstraction from the matter ” (1956: 1; author’s emphasis). Thus,
for Church the science of logic is a metalogical study of underlying logics (1956:
57-58). The difference between logic and metalogic is drawn between using a
logic to process information about a given subject matter with a given object
language and studying a logic or an underlying logic, which involves a language,
a semantics, and a deduction system. Logicians use a metalanguage to study
the formal aspects of an object language apart from its subject matter, often to
study an underlying logic’s deduction system. Aristotle undertook just such a
study in Prior Analytics. Indeed, part of Aristotle’s philosophical genius is to
Aristotle’s Underlying Logic
107
have established a formal logic, while at the same time making the study of logic
scientific. He recognized that deductions about a given subject matter are topic
specific and pertain to a given domain, say to geometry or to arithmetic or to
biology, but that such deductions employ a topic neutral deduction system to
establish knowledge of logical consequence.
In having a keen interest in epistemics, Aristotle shares with modern logicians
the notion that central to the study of logic is examining the formal conditions
for establishing knowledge of logical consequence — that logic, then, is a part of
epistemology. He composed Prior Analytics and Posterior Analytics to establish a
firm theoretical and methodological foundation for a7to§EixTLxr) ejucrrrjjjr] ( apodeik -
tike episteme ), or demonstrative knowledge (24al0-ll). In Nicomachean Ethics
6 , where he treated the intellectual excellences, Aristotle indicated the importance
he attributed to demonstration (omo&ei&c; [apodeixis]): “scientific knowledge, then,
is a demonstrative state omoSeocuxV)]” that constitutes an appropriate confi¬
dence in the results of deductive reasoning (1139bl8-36; cf. Po.An.A2 : 71bl8-22).
He saw his purpose in Prior Analytics precisely to establish confidence in the de¬
duction process and particularly in his syllogistic deduction system. To accomplish
this project he especially studied the formal or syntactic matter of deducibility.
Aristotle thought of deduction as a kind of computational process. Indeed, the
verb ‘auX^oyiCeodoa’ (sullogizesthai ) used by Aristotle to denote the special kind
of deduction process treated in Prior Analytics derives from mathematical calcu¬
lation. His special concern, then, was to develop a deduction apparatus by which
someone could decide in a strictly mechanical , or computational, manner which
sentences are logical consequences of other sentences.
Aristotle’s promethean contribution to science and philosophy, then, concerns
his study of the deduction process itself. He knew that a given sentence is either
true or false; and he recognized this to be the case independent of a participant.
He also knew from his familiarity with mathematical argumentation and dialec¬
tical reasoning that a given sentence either follows necessarily or does not follow
necessarily from other given sentences; likewise, he recognized this to be the case
independent of a participant. These are ontic matters having to do with being.
In addition, Aristotle knew that the truth or falsity of a given sentence or the
validity or invalidity of a given argument might not be known to one or another
participant. Now, a given axiomatic science aims to establish knowledge about its
proper subject matter ( Po. An. Al: 71al-ll & A3: 72bl9—22). Since Aristotle
took such a science to consist principally in the collection of sentences — defini¬
tions, axioms, theorems — of its extended discourse, the project of such a science
is to decide which sentences pertaining to its subject matter are true, or theorems,
and which sentences, for that matter, are false and not theorems. Procedures
for deciding a sentence’s truth or falsity are epistemic matters having to do with
knowing.
In respect of epistemics Aristotle recognized two ways to establish the truth of
a given sentence: (1) by induction (eTtaYwyrj) and (2) by deduction ( Po.An.Al-2 ,
EN 6.3 & Meta. 1.9: 992b30-993al). In respect of an axiomatic science, while
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George Boger
definitions and axioms, or first principles, are determined inductively and are not
the result of a deductive process, 4 its theorems are decided deductively. In the
works of the Organon , particularly in Prior Analytics and Posterior Analytics,
Aristotle treated the deductive method for establishing knowledge that a given
sentence is true. This project requires two steps (Pr. An. Al: '25b28 31), which
he treated separately in Prior Analytics and Posterior Analytics. In Posterior An¬
alytics Aristotle treated the requirements for demonstrative science, a constituent
part of which is demonstration. He writes in Posterior Analytics A2:
By a demonstration [omohedfiv] I mean a scientific deduction [auXXo-
ytapov £Tuaxr)povLx6v]; and by scientific I mean a deduction by possess¬
ing which we understand something ... demonstrative understanding
[xf)v dutoSeiXTixqv Ejuarf)pr)v] in particular must proceed from items
that are true and primitive and immediate and more familiar than and
prior to and explanatory of the conclusions. There can be a deduction
[auXXoytapo? (sullogismos) Jeven if these conditions are not met, but
there cannot be a demonstration [dnt68ei£i<;] — for it will not bring
about understanding [eiuaifjprjv]; in respect of a given subject matter].
(71bl7-25; cf. Top. Al: 100a27-29)
Aristotle early distinguished deduction ( sullogismos ) from demonstration
(apodeixis ). In Prior Analytics Af he stated that he would treat deduction before
demonstration because it is more universal: “for [every] demonstration is a kind of
deduction, but not every deduction is a demonstration” (25b30—31). In Posterior
Analytics A2 (cf. Pr. An. B2~4) he determined this universality to consist in a
deduction’s being possible even when the premiss sentences are not antecedently
known to be true or even when they are false. Thus, one can know that the con¬
clusion sentence of a given demonstration is true because (1) its premiss sentences
are all true and (2) it is a deduction.
A deduction per se, then, establishes knowledge, not that the sentence that is
the conclusion of a given argument is true, but only that it follows necessarily, or
logically, from the sentences in a premiss-set. Aristotle made an important distinc¬
tion in his logical investigations between epistemic concerns and ontic concerns.
This is especially evident in Prior Analytics Bl-f where he treated the deducibil¬
ity of true and false sentences from various combinations of true and false sentences
taken as premisses. This distinction indicates an understanding of logical conse¬
quence that modern logicians will recognize. The confidence one acquires from a
demonstration derives from knowing, as Aristotle often pointed out, that it is im¬
possible for true sentences to imply a false sentence (Pr. An. B2-f). Given true
4 A science’s first principles are part of the premiss-set in an extended deductive discourse.
Aristotle referred to these as “unmiddled” (apeaov), immediate or indemonstrable, that is, not
themselves products of demonstration; their truth is established independently. See Po.An.A3
on the notion that not every categorical sentence is demonstrable. In an extended deductive
discourse the derived theorems are added to the original principles as additional premisses for
subsequent derivations. Euclid’s Elements serves as an ancient example.
Aristotle’s Underlying Logic
109
sentences as premisses, established (initially) by means independent of deduction,
one can be certain that the conclusion sentence of a demonstration also is true 5
precisely because it is shown to be a logical consequence of other true sentences. In
Prior Analytics Aristotle was especially concerned to determine which formal pat¬
terns of argumentation might be used to establish knowledge that a given sentence
necessarily follows from other given sentences. In particular, he saw his project as
determining “how every syllogism is generated” (25b26- 31) by identifying which
elementary argument patterns could serve as rules analogous to such patterns as
modus ponens, modus tollens , and disjunctive syllogism for modern propositional
logic.
Looking back, we see that mathematicians of the fourth century had been as¬
siduously attending to axiomatizing geometry. This activity principally concerned
condensing the entire wealth of geometric knowledge into small sets of definitions
and axioms from which the theorems of geometry could be derived and set out as
a long, extended discourse. Euclid’s Elements is an extant fruit of this activity.
Except for identifying a small set of common notions (xotvai apxcd or ra xoiva),
these mathematicians were not concerned with studying the epistemic process
underlying geometric discourse. They took geometry intuitively as an informal
axiomatic system (Church 1956: 57) with an implicit underlying logic. The an¬
cient mathematicians may have formalized the truths of geometry, but they hardly
formalized the deductive method for processing the information already contained
in its definitions and axioms.
Undoubtedly Aristotle had participated in discussions, in the Academy and
elsewhere, about axiomatizing geometry. He may have asked about deduction
rules used to establish geometric theorems. Indications that he did include his
attention to various proofs such as that of the incommensurability of the diagonal
with the side of a square and those related to properties of triangles, and his
frequent attention to the common notions of the mathematical sciences; there is
also his curious mention of the middle term and syllogistic reasoning in connection
with geometric demonstration (Po. An. A9: 76a4-10; A12: 77b27-28; cf. Pr. An.
A35 & A24). Aristotle surely wondered how one could be assured in geometric
demonstrations that a conclusion necessarily follows from premisses. This matter
is all the more interesting in the case of longer, more involved demonstrations.
Still, we cannot say that he undertook a metalogical study of geometric proof.
Perhaps it was Aristotle’s own insight or an implicit part of the philosophical
discussion of the time that the axiomatization of geometry could serve in some
way as a model for formalizing the non-mathematical sciences such as botany
and zoology. Some such notion seems to have animated his scientific and logical
investigations. 6 Now, the actual project of establishing a given science’s definitions
5 Aristotle recognized two necessities in a demonstration when he remarked about that which
“it is impossible for it to be otherwise” (Po. An. A2: 71bl5-16): (1) that having to do with
the subject matter of a demonstration — that about which; (2) that having to do with logical
consequence — that following necessarily. See below Section 5.3 on Aristotle’s notion of logical
consequence.
6 See, for example, on the matter of Aristotle’s application, or lack of application, of his
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George Boger
and axioms and then its theorems, which would conform to the criteria set out in
Posterior Analytics A2 , did not concern Aristotle in the Organon. This project
lies outside the scope of logic. Rather, his preeminent concern there was to study
deduction and demonstration per se: not with, that is, one or another distinct
subject matter, but with the formal deduction process that has no similar subject
matter. Aristotle in his logical investigations subordinated a concern with the what
or the why and wherefore to focus on the that and the how. Thus, presupposing
various axiomatic sciences with distinct domains, he took up the narrower and
more poignant questions about the epistemic process of deriving theorems from
axioms. Aristotle especially examined the deductive foundations of demonstration,
that is, of demonstrative knowledge or axiomatic science.
The first chapters of Metaphysics 1 reveal Aristotle’s intellectual disposition
toward scientific knowledge and signal the importance he attributed to metalogi-
cal study of deduction. In fact, Aristotle identified this task as a province proper
only to philosophy. In Metaphysics 2.1 he writes that “philosophy is the science of
truth [t j]v tpiXoaotplav eKiaifjpqv xf)p aXrjGdap]” (993b20). And since “status in be¬
ing governs status in truth [cooO’ exotaxov ox; e'xei tou eivou, outco xai Tfjc; dXr}0ELac;]”
(Meta. 2.1 : 993b30—31), the philosopher’s project includes studying, not a par¬
ticular part of being, but being-gua-being (to ov f) ov; Meta, f.l: 1003a21-26).
“The philosopher must have within his province the first principles [ia<; apxap] and
primary factors of primary beings” (Meta, f.2: 1003bl7—19) and “be able to view
things in a total way” (1004a34-1004bl). Accordingly, “it is not the geometer’s
[nor any other specialist’s] business to answer questions about what contrariety is,
or perfection, or being, or unity, or sameness, or diversity [or even, for that mat¬
ter, about deduction rules]; for him these remain postulates [e£ utcoOegeox;]” (Meta.
4-2: 1005all-13; cf. 1005a31). Later in Metaphysics 4-3 Aristotle addressed the
philosopher’s responsibility to examine certain axioms precisely because they refer
to all of being — being-gwa-being — and not just a part of being (1005a21-22).
But it is clear that the axioms extend to all things as being (since they
all have being in common); hence the theory of axioms [rcepl toutcov
(a^lwpaTOJv) eaxlv f) Gswpla] also belongs to him who knows being as
being. (1005a27-29)
He is not writing about axioms special to a particular science, but about ontic
principles that apply alike to all domains.
After indicating the limitations of the special sciences for examining these ax¬
ioms, Aristotle writes that
the philosopher, who examines the most general features of primary
being, must investigate also the principles of deductive reasoning [twv
auXXoyi.cmxcov otpyciv]. ... So that he who gets the best grasp of be¬
ings as beings must be able to discuss the basic principles of all being
syllogistic to the actual project of axiomatizing a natural science: J. Lennox 1987, R. Bolton
1987, J. Hintikka 1996, W. Wians 1996, D. K. W. Modrak 1996, J. J. Cleary 1996, and Graham
1987.
Aristotle’s Underlying Logic
111
[tac itavTcov [3epaioxdtxotc;], and he is the philosopher. (1005b5-6-ll; cf.
Meta. 11-4)
Immediately Aristotle cites the principle of non-contradiction as one of the princi¬
ples “about which it is impossible to be mistaken” and writes, moreover, that such
a principle that is “[necessary] in order to understand anything whatever cannot
be an assumption [xouxo oux U7t60£ai<;]” (1005bll-12 & 15-16): “It is impossible
for the same thing at the same time to belong and not to belong [uixapysiv xe
xoti pf] OxdpxcLv] to the same thing and in the same respect” (1005bl9-22). He
states the principle here as an ontic principle, 7 but he immediately relates it to
demonstration, implicitly reminding us that being governs truth.
Hence, if contraries cannot at the same time belong to the same thing
and if an opinion [8o£a] stated in opposition to another opinion
is directly contrary to it, then it is evidently absurd for the same man
at the same time to believe the same thing to be and not to be; for
whoever denies this would at the same time hold contrary opinions
[xot? Evavxtap 8o£a<;]. It is for this reason that all who carry out a
demonstration rest it on this as on an ultimate belief [Ecrx“ Tr l v &o£av];
for this is naturally a foundation also of all other axioms [cpuasi yap
apxfj xai xwv aXXwv d&upaxuv auxr) xavxcov], (1005b26-34)
This passage ends Metaphysics 4-3. Much of the remainder of Metaphysics 4 is de¬
voted to establishing the absurdity of rejecting the principle of non-contradiction
for intelligible discourse. It is evident that Aristotle understood a philosopher’s
responsibilities to include examining the principles of deductive reasoning. For
Aristotle, studying the principles of being is simultaneously a study of the princi¬
ples of thought. Logic, which he took to be a part of epistemology, is nevertheless
grounded in the nature of being. Perhaps Aristotle appropriated Parmenides’ dic¬
tum that “thought and being are the same [xo ydp auxo voelv fe'cmv xs xai sivai]”,
taking this to mean that truth and logical consequence, as they appear in thought,
follow being.
Now, Aristotle writes in Metaphysics 11.4 that “taking equals from equals leaves
equal remainders” is an example of a notion common to all quantitative being
(1061b20-21). He had recognized that the common notions belonging to the math¬
ematical sciences belonged equally to all and specially to none. 8 And, although
7 Aristotle provides both ontic and epistemic expressions of this principle in Meta. 4 ■ Here he
provides an ontic expression, but in Meta. 4-4 he expresses this law in the following way: “If
when an affirmation [cpacru;] is true its denial [djt 09 aoii;] is false, or when the denial is true its
affirmation [xaidcpaau;] false, then it will not be possible at the same time to assert [qxxvtxi] and
to deny [dxocpdvai] the same thing truly” (1008a34-1008bl). See §5.3.
8 Consider Po. An. A5: 74al7-25. “Again, it might be thought that proportion alternates
for items as numbers and as lines and as solids and as times. In the past this used to be proved
separately, although it is possible to prove it of all cases by a single demonstration: because
all these items — numbers, lengths, times, solids — do not constitute a single named item and
differ in form from one another, they used to be taken separately. Now, however, it is proved
112
George Boger
he never articulated a complete list of common notions for the non-mathematical
sciences, nor for that matter in any systematic way for the mathematical sciences,
he stated in Metaphysics 4 that the principles of contradiction and of the excluded
middle are among the common notions applicable to all rational discourse. Aristo¬
tle noticed that the common notions relating to different branches of mathematics,
while stipulative of magnitudes in general, do not stipulate any one domain, such
as arithmetic or geometry, in particular. They generally do not specify any content
however much they anticipate establishing relationships among magnitudes special
to a mathematical science. They are topic neutral in this circumscribed sense, and,
thus, they have a relative independence unlike lines, angles, or numbers. They are
neither embedded in nor mentally inextricable from the objects of a particular
science. This is not the case with a science’s principles. The common notions,
then, may be taken in abstractum and treated on their own account irrespective
of the subject matter of a given quantitative science.
These common notions, moreover, generally express formal relationships among
magnitudes within a quantitative science and accordingly apply equally to a vari¬
ety of different, quantitative domains. Because of their relative formality and their
special universality, the common notions were applied as inference rules across the
mathematical sciences. Their use in this epistemic manner is evident in Euclid’s
Elements. 9 Aristotle’s having recognized the common notions as principles of
reasoning had profound consequences for the development of ancient logic. Un¬
derstanding this and that Aristotle took a syllogism to fit an elementary argument
pattern with only valid instances help to confirm the rule-nature of his statements
in Prior Analytics A4-6 relating to when a syllogism comes about. In the case
of Euclid’s common notions, two magnitudes remain incommensurable without
there being a third, or middle, that unites them as extremes in a particular way
— using a common notion makes this evident. An exactly analogous relationship
applies in the case of the patterns of perfect or complete syllogisms, the teXeioi
cruXXoYtapoi (teleioi sullogismoi) , in Prior Analytics A4 in respect of relating
substantive terms and making evident their connections. Aristotle’s manner of
expressing the patterns of the syllogisms in sentences beginning with ‘eav’ and ‘el’
comports exactly with Euclid’s expressions of the common notions and suggests
their similar rule-nature. 10 Aristotle went on to express his rules using schematic
letters where Euclid did not. Scholars have overlooked Aristotle’s written state¬
ments of the rules to see only their schematic representations. Consequently, they
have not recognized this as part of his effort to model a logic. Accordingly, they
universally: what they suppose to hold of them universally does not hold of them as lines or
as numbers but as this" We use J. Barnes’s (1994) translation of Posterior Analytics with
modification.
9 I. Mueller (1981: 32-38), following T. Heath (1956: 117-124, 221-226), maintains that
Euclid’s common notions serve as deduction rules in his proofs. He believes this is the case even
allowing that Euclid did not treat the matter of deducibility. Cf. B. Einarson 1936: 42-49 and
H. D. P. Lee 1935: 114-115.
10 See T. Heath 1956: 120-122 on ancient ideas of the common notions. Also see Heath 1956:
221-232 on Euclid’s treatment of the common notions.
Aristotle’s Underlying Logic
113
missed this link to mathematics and thus they missed an important part of his
theory of deduction.
Finally in this connection, besides stating syllogistic deduction rules and his
actually using the patterns of the teleioi sullogismoi as rules in Prior Analytics,
Aristotle virtually stated his taking them formally as rules in Prior Analytics A30
(46al0-12/15). There he used the expression ‘the principles of deduction’ — “on
tuv auXXoyiapwv otp)(o d” — that we also encounter in Metaphysics 4-3 (1005b7).
Aristotle did not refer here in Prior Analytics to the principles or axioms of a
given science, but to the most general principles of all being as they are grasped in
thought. And again, in this same connection, he used the expression ‘the principles
of demonstrations’ — “on xwv aTto&eixxixwv apxod” — in Metaphysics 3.2 (996b26).
Indeed, throughout Metaphysics Aristotle used the following expressions as syn¬
onyms in referring to common notions, including the laws of non-contradiction and
the excluded middle: ‘xa xotva’ (1061bl8), ‘apyod’ (996b26, 997al3), ‘d^icapaxa’
(997all, 13), and ‘xoivai So^ai’ (996b28, 997a21; see esp. Po. An. A10-11).
Thus, we can see that a pattern of a syllogism is a relatively uninterpreted object.
In fact, Aristotle treated each pattern exactly as a topic neutral rule of deduction
in Prior Analytics A^-7analogous to Euclid’s use of common notions in Elements.
Perhaps the patterns of the four teleioi sullogismoi are Aristotle’s adaptation to
the non-mathematical sciences of the common notions employed as deduction rules
in the mathematical sciences.
1-4 The scope of this study
Our concern here is to present Aristotle’s system of logic while also revealing the
mathematical sophistication of his logical investigations. Modern logicians believe
that the possibility of mathematical logic, an important part of which involves
generating models, consists in making a clear distinction between syntax and se¬
mantics. They also believe that the clear distinction between syntax and semantics
resulted from borrowing symbolic notations from mathematical practice and then
applying them to studies of deductive logics, but that earlier thinkers, lacking such
notations, could not have made such distinctions. However, mathematical logic,
considered as a discipline in general, has a formal and a material aspect. Its for¬
mal aspect has principally to do with the symbolic notations that have helped to
illuminate underlying structural, or logical, features of deductive discourses. Yet,
the substance of mathematical logic does not consist in its sophisticated notations,
but in the problems logicians consider when studying underlying logics — that is,
in particular, w'hen they distinguish a logic’s syntax and its semantics and then ask
questions about their relationships. Principal in this respect have been questions
about a logic’s consistency, soundness, and completeness, which involve determin¬
ing relationships between deducibility and logical consequence. A distinguishing
feature of mathematical logic, then, consists precisely in these substantive mat-
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George Boger
ters. 11 Remarkably, with only a rudimentary notation Aristotle considered just
such mathematical matters in his concern to establish the practical, epistemic
power of his logic for establishing scientific knowledge. In this connection we can
grasp the revolution in the history and philosophy of logic — the “hypostatiza-
tion of proof’ — consolidated by Aristotle’s works on logic. “Prior Analytics is
the earliest known work which treats proofs as timeless abstractions amenable to
investigation similar to the investigations already directed toward numbers and
geometrical figures” (Corcoran: personal communication). Thus, Prior Analytics
is a proof-theoretic treatise on the deduction system of an underlying logic. Aris¬
totle recognized the epistemic efficacy of certain elementary argument patterns, to
wit, those of the syllogisms, and he formulated them as rules of natural deduc¬
tion. Having raised important metalogical questions about the properties of his
syllogistic deduction system, he successfully established a set of formal, epistemic
conditions for recognizing logical necessity, and in this way he became the founder
of formal logic.
Below we set out Aristotle’s underlying logic much as he himself did in the
works of the Organon. We include Metaphysics among the treatises of his logical
investigations. It is natural and not surprising that modern logicians and commen¬
tators, when treating Aristotle’s logic, focus principally on Prior Analytics: Prior
Analytics is the most ‘logical’ of the treatises. In truth, the attraction of Prior An¬
alytics has consisted in a scholar’s implicit recognition that Aristotle there treated
the deduction system of an ancient underlying logic. We say ‘implicit’ because
it was not until the studies of J. Corcoran and T. Smiley, and later those of R.
Smith, that there is a growing explicit recognition that this is so. In any case, a
deduction system is only one part of an underlying logic, which also contains a
11 While the initial impetus of modern logic involved axiomatizing geometry and number theory
and attempting to reduce mathematics to logic, it was David Hilbert in the 1920s who turned
attention specifically to the formal deduction process and made it an object of mathematical
investigation. He worked on devising an algorithm or decision procedure. Moreover, he noted
that the semantic concepts of validity and satisfiability coincide with the syntactic concepts of
derivability and consistency. Hilbert emphasized the study of such syntactic questions as those
of consistency and completeness, which he considered to fall under what he called “metamath¬
ematics”, or “proof theory”. We take this attention on deduction systems to be a substantive
concern of mathematical logic, whatever formal systems mathematical logicians may devise. Of
course, modern mathematical logic far surpasses the accomplishments of Aristotle, particularly
in respect of devising and studying uninterpreted formal systems. We need only mention here
the developments of genuine variables, functions, quantification, set theory, highly formal lan¬
guages, recursion theory, and model theory. Nevertheless, if we take logic principally to treat
underlying logics as A. Church, then we see that mathematical logic fundamentally addresses
foundational questions about deduction systems. In addition, the origin of modern mathematical
logic involved axiomatic systems, both in respect of ontic subject matter and epistemic formal
matters. This was just the kind of concern that occupied Aristotle himself, whom we detect as
distinguishing the content of an axiomatic science from the deduction apparatus used to establish
its theorems. Aristotle came to address similar foundational questions, although likely he did not
come to them as modern logicians initially did by way of attempting to reduce mathematics to
logic. On Aristotle’s having proof-theoretic interests, see R. Smith 1984: 594-596, 1986: 55-61,
and 1991: 48-50. J. Corcoran (1974, 1994) and R. Smith (1989) have generally made Aristotle’s
case in this respect.
Aristotle’s Underlying Logic
115
grammar and a semantics. Our contribution takes this recognition a little farther
to hold that Aristotle intentionally aimed to develop an underlying logic along the
lines of modernist thinking. This means that Aristotle invented a formal language
to model his logic. However, since Aristotle did not set out his underlying logic in
as systematic a manner as a modern logician, while, nevertheless, accomplishing
much the same result, we employ the theoretical apparatus of modern mathe¬
matical logic to structure his account. With the aid of this template we show in
Aristotle’s own words that he was concerned with exactly similar matters as a
modern logician. We begin by presenting Aristotle’s treatment of the syntax and
semantics of natural language in Categories, On Interpretation, and Metaphysics.
These studies laid a foundation for his developing the formal language found in
Prior Analytics for modeling axiomatic discourse. We then proceed to extract the
syntax of sentence transformations leading to his establishing a set of deduction
rules in Prior Analytics. Next we treat the logical methodology by which Aristotle
established his deduction rules. We conclude with a statement of his understand¬
ings of “formal deducibility” and “logical consequence” and with a final section
that summarizes four proof-theoretic accomplishments of his logical investigations.
1.5 Logic terminology
The following terminology assists in our study of Aristotle’s logic. We use Aris¬
totle’s own terminology wherever it exists, which, interestingly, often corresponds
exactly to ours. An argument is a two part system consisting in a set of sentences
in the role of premisses and a single sentence in the role of conclusion; an argu¬
ment is either valid or invalid. A sentence is either true or false. We sometimes
use ‘conclusion’ elliptically for ‘sentence in the role of conclusion’ or ‘conclusion
sentence’, and similarly for ‘premiss’. An argumentation is a three part system
consisting in a chain of reasoning in addition to premisses and conclusion; an argu¬
mentation is either cogent, in which case it is a deduction, or fallacious, a fallacy.
A sentence, an argument, and an argumentation are object language phenomena
and domain specific. An argument pattern is a two part system consisting in a
set of sentence patterns in the role of a premiss-set and a single sentence pattern in
the role of a conclusion. A pattern is a metalinguistic object distinguishable from
a form and is commonly represented schematically. An argument is said to fit, or
to be an instance of, one or more argument patterns . 12 A given argument pattern
may have all valid instances, all invalid instances, or some valid and some invalid
instances. An argument pattern is not properly valid or invalid, although logicians
have used ‘valid’ in this connection, but we distinguish these category differences.
An argument pattern with all valid instances is panvalid, that with all invalid
12 We use “pattern” and “form”, following J. Corcoran (1993: xxxi-xxxvii) as, e.g., Irving
Copi (1986: 288-291) uses “form” and “specific form” and as Willard O. Quine (1982: 44) uses
“general [logical] schema” and “special case [logical schema]”. Cf. W. O. Quine 1970: 47-51.
We can express this difference in the following way: while a given argument has only one form,
it might fit more than one pattern. See also Corcoran 1989.
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George Boger
instances is paninvalid, and that having instances of both is neutrovalid. We
add that an elementary panvalid argument pattern is one having a simple
premiss-set pattern whose epistemic value consists, in many cases, in its being
quickly evident, or ‘evident through itself’, that its conclusion follows necessarily.
An elementary argument pattern may be formulated in a corresponding sentence
to express a rule of deduction. In addition, we follow G. Patzig (1968; cf. Rose
1968) to distinguish in Aristotle’s logic a concludent pattern of two sentences in
the role of premisses, or a premiss-pair pattern, from an inconcludent premiss-
pair pattern. A concludent pattern has a necessary result, that is, it results in
a panvalid pattern all of whose instances are syllogisms, while an inconcludent
pattern has no necessary result, that is, it cannot result in a panvalid pattern but
only in a paninvalid pattern with only invalid argument instances.
In addition, we understand Aristotle to have considered a auXXoyi.ap6<; ( sullo-
gismos ), which we translate by ‘syllogism’, to be a valid argument with only two
premiss sentences, having only three terms, in one of three figures. 13 A syllo¬
gism, then, is an elementary argument fitting a panvalid pattern. No syllogism is
invalid. Aristotle saw his project in Prior Analytics to identify all such patterns,
precisely because of their epistemic efficacy in the deduction process. We use the
traditional names of the ‘syllogisms’ — ‘Barbara’, ‘Celarent’, etc. 14 — to name
patterns of syllogisms, just as ‘ modus ponens ’ names a kind of familiar pattern in
propositional logic used in a deduction process. Still, these names do not signify
instances of such patterns. Of course, ‘Barbara’ and ‘modus ponens ’ also name de¬
duction rules. However, in some cases — especially those pertaining to the teleioi
sullogismoi , those in Sophistical Refutations , and those in Prior Analytics when
Aristotle refers to a sullogismos as proving something — we translate ‘ sullogismos’
by ‘deduction’. In these cases Aristotle recognized an epistemic process to occur
in the mind of a participant who grasps that a given sentence is a logical conse¬
quence of other given sentences. Still, when he writes, in relation to a deduction
process, that a syllogism arises (yivexai), we understand him not to mean that a
syllogism per se is a deduction, but that one’s arising during a deductive chain of
reasoning signals making logical consequence evident, just as when a participant
links given propositions and produces an instance of modus ponens signals logical
consequence in propositional logic.
Finally, we take treating patterns of sentences, patterns of arguments, and pat¬
terns of argumentations to constitute a large part of modeling a logic. Thus, for
13 In Boger 1998 I used ‘sullogismos’ rather than translate ‘aoXXorLairoq’ better to objectify
its meaning. I argued that a sullogismos, as Aristotle treats it in Pr. An. AJ t ~l, 23, h 45, is
a panvalid argument pattern and neither an argument nor a deduction. However, Prof. David
Hitchcock of McMaster University, while agreeing that Aristotle treated patterns in Pr. An., has
convinced me that he did not call them ‘sullogismoi’ but reserved the word for valid arguments
fitting such patterns.
14 The traditional names of the syllogisms in the first figure are Barbara, Celarent, Darii, and
Ferio. The traditional names of second figure syllogisms are Camestres, Cesare, Festino, and
Baroco and of the third figure Darapti, Datisi, Disamis, Felapton, Ferison, and Bocardo. See W.
T. Parry (1991: 282) for a short explanation of their origin and meaning.
Aristotle’s Underlying Logic
117
example, while there are numerous simple sentences in a given object language,
each of them, nevertheless, consists in a subject and a predicate. Extracting this
elementary pattern and representing it abstractly, or metalinguistically, is model¬
ing a simple sentence — either by means of another sentence, using the language
of the given object language (but, nevertheless, in the metalanguage), or by means
of mathematical notation. In either case, a sentence is modeled and becomes an
object of logical investigation. Thus, we take a formal language to be a model of
one or another object language, with one or another degree of precision. In this
way a logician can model arguments, deductions, and deduction systems better to
study their respective properties and logical relationships.
2 ARISTOTLE’S ANCIENT MODEL OF AN UNDERLYING LOGIC
Aristotle knew that deductions about geometric objects are topic specific and that
they employ a topic neutral deduction system, even if a participant uses that sys¬
tem implicitly. In Prior Analytics he turned his attention not to geometric or
biological objects, nor even to geometric or biological discourses, but to the deduc¬
tion apparatus used to make evident that a given categorical sentence necessarily
follows from other given categorical sentences. 15 Aristotle had observed a num¬
ber of elementary argument patterns used in various object language discourses,
some of which he recognized in their use always to result in something following
necessarily, others of which he recognized in their use never to result in something
following necessarily. He subsequently extracted these patterns for systematic ex¬
amination. In Prior Analytics Aristotle modeled his syllogistic logic and presented
the results of his investigating these patterns. In this connection, then, Prior Ana¬
lytics is a scientific study of the syllogistic deduction system, which, taken together
with Categories , On Interpretation , and parts of Metaphysics, comprises Aristotle’s
treatment of an underlying logic.
The logic underlying cogent object language discourse accounts for that dis¬
course’s coherence. While this discourse is itself topic specific as it treats objects
of a given domain, its underlying logic is topic neutral and not bound to any one
subject matter. The science of logic is devoted in great measure to modeling these
underlying logics and consists in their study. To accomplish this study, a logician
must not only model the deduction system of such discourse, but he/she must also
model the object language itself, often with an aim to make such a language more
precise. A logician’s principal concern is to extract and formalize (1) a grammar
for the formation of sentences and their relationships and (2) a deduction system
for sentence transformations. These are formal, syntactic concerns. A logician
constructs a formal language to model one or more object language in respect
of its structure. Such a formal language is taken to be uninterpreted, although
15 While Aristotle named the parts of a categorical sentence — subject term and predicate
terms, and the logical constants — he did not use the expression ‘categorical sentence’. This
expression is a later invention that nevertheless captures his meaning and distinguishes this kind
of sentence from those of, for example, natural Greek, or natural language.
118
George Boger
hardly is such a formal language purely uninterpreted — often its logical constants
are interpreted or have an implicit intended interpretation, as are what count as
a sentence and an argument, etc. In any case, the ‘formulas’ or patterns for con¬
structing and transforming sentences are relatively uninterpreted, as evidenced
by the impossibility of assigning them meanings and truth-values (save for logics
with identity and tautology). Thus, in modeling an underlying logic a logician also
treats the semantics of sentences — establishing meaning and truth conditions —
and of sentence transformations — establishing conditions of logical consequence.
Aristotle, then, invented an artificial language for two closely related purposes
that embrace a modernist concern for modeling a logic. First , he wanted to develop
a language (1) that conformed to his ontology of substance, a core of which is
presented in Categories, and (2) that promoted a precision in scientific knowledge,
a concern that he forcefully expressed in Metaphysics. Second, he wanted to model
the underlying logic he developed as an epistemic instrument for scientific discourse
both (1) to facilitate determining the properties of his logic and (2) to represent
his logic for instructing others in its use. It is doubtful that Aristotle developed
this artificial language to model natural language and more likely that he aimed
to standardize scientific discourse and to model his logic. Aristotle invented four
categorical sentence patterns, and he treated them as formal objects in order to
establish certain of their properties and logical relationships. And while he did
not represent his logic with a modern rigor and system, we can easily organize
his own discourses according to a mathematical template without distortion to his
meaning and intention. In this section we first consider Aristotle’s treatment of
the grammars of natural language and his artificial language (§2.1), second, the
semantics of his language (§2.2), and, third, the syntax of his deduction system
(§2.3). While Aristotle treated the syntax of sentences in close relation to their
semantics, he nevertheless sufficiently distinguished them so that we can treat
them separately.
2.1 Aristotle’s metalinguistic study of grammar
To extract and represent his deduction system for analysis in Prior Analytics, and
to prepare for its application to the various axiomatic sciences as a science is con¬
strued in Posterior Analytics, Aristotle undertook a systematic study of language.
While the Sophists are perhaps his more immediate predecessors in this connection,
Aristotle’s contributions firmly consolidated the early stages of linguistics as a spe¬
cial branch of learning. Efforts in this area are especially evident in Categories, On
Interpretation, Metaphysics, Topics, Sophistical Refutations, and Rhetoric. In On
Interpretation Aristotle treated the complexity of Greek grammar only generally
as suited the purpose of his logical investigations. There he identified the simple
sentence that predicates one thing of another thing as a proper object of logical
analysis. By studying a natural language in these treatises Aristotle prepared
the way to inventing the artificial language in Prior Analytics, perhaps the first
Aristotle’s Underlying Logic
119
artificial, or formal, language in the history of philosophy. And he accomplished
this task without the aid of a sophisticated system of symbolic or mathematical
notation. With his treatment of predication in Categories and Metaphysics in the
background, 16 we turn now to a part of the elementary grammar examined in On
Interpretation.
The grammar of a natural language
In On Interpretation 1-f Aristotle writes about sentence formation in a natural
language, in this case in his own natural language. There he uses Greek to mention
and to illustrate his observations about intelligible discourse that might apply in
principle to any language (16a5-6). In this connection he intuitively takes Greek to
be what modern logicians call a fully interpreted language. Nevertheless, he care¬
fully focuses attention on its structural aspects apart from any meanings, except
for purposes of illustration, that sentences might express about a subject mat¬
ter. Indeed, although he does not have expressions for ‘natural language’, ‘object
language’, and ‘metalanguage’, it is evident that in On Interpretation Aristotle
intentionally objectifies aspects of language in general and does not study only
the Greek language in particular. On Interpretation is a metalinguistic treatise
in which Aristotle consciously examines certain syntactic and semantic aspects of
language.
Aristotle treats sentence formation in a natural language as essentially consist¬
ing in combining (auvQeaic) a noun (ovoua) and a verb i.e., a predicate
[16a9-18]) so as to produce a meaningful expression (cptovr) arpavcixf)), a complete
thought. Every sentence necessarily has these two basic components, neither of
which by itself is sufficient.
Every affirmative sentence [iraoa xaxdcpaau;] consists in a noun and a
verb, whether [determinate or] indeterminate. Unless there is also a
verb, there is neither an affirmation nor a denial [oOSepla xortacpaaic;
ou5’ otTtocpotcTu;] . (On Int. 10: 19blO—12; cf. Cat. 2: lal6-19) 17
In addition, Aristotle recognizes that the words making up a sentence must be
concatenated or strung together in certain ways so as to bear meaning: “that
16 Perhaps Categories (Katrjyoptai) would better be named “On predication” or “On predi¬
cating properties of substance”. Categories is a metalinguistic treatise on sentence formation,
which, moreover, aims at precision and truth in the sciences. This treatise avows Aristotle’s
subscription to a correspondence notion of truth, as does On Interpretation as well as to a mate¬
rialist theory of nature, or substance. While much has been said about Categories, we here only
provide a brief reference to Aristotle’s theory of substance. Of the ten categories (listed in Cat.
4) — viz. substance (oumoc), quantity (tiooov), quality (tioiov), relation (npoi; ti), place (itou),
time (^oxe)i position (xelaOai), condition (e'xsuv), action (noieiv), and affection (jca o/eiv) — sub¬
stance is predicated of nothing but the others are predicated of a substance, a ‘this’. Substance
is treated in Cat. 5, quantity in Cat. 6, relation in Cat. 7, quality in Cat. 8, and the others in
Cat. 9.
17 For translations of Categories and On Interpretation we work with H. P. Cooke’s (1938) and
J. L. Ackrill’s (1984) translations and make significant modifications, e.g., translating ‘ditotpavou;’
by ‘sentence’ and by neither ‘proposition’ as Cooke nor ‘statement’ as Ackrill.
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George Boger
the words are pronounced [merely] in succession [auveyyuc;] ] does not make them
a unity [ere;]” (17al4). 18 In On Interpretation 4 Aristotle defines ‘sentence’ as
follows:
A sentence [Xoyoc;] is meaningful speech [epeovrj arjpavxtxfj] — the parts
of which, as expressed [separately] mean something as an expression
but not as an affirmation [ox; cpdou;, otAX’ oG)( dx; xaxdcpaau;]. I mean,
for example, that ‘man’ means something, but [by itself] not that it is
or is not ; there will be an affirmation or a denial [only] if something is
added [xi xpoaxeGf)]. (16b26-28; cf. 10: 19blO-T2)
A noun and a verb by themselves may possess meaning, but by themselves they
do not constitute a sentence, nor do they constitute a sentence merely by being
strung together arbitrarily. Thus, from his notion of sentence in On Interpretation,
we can extract Aristotle’s rule for the formation of a generic sentence in a natural
language and express it as follows:
SFR1 A sentence in a given natural language consists in combining a noun and
a verb (i.e., a predicate) in certain ways so as to produce a meaningful
expression.
This rule identifies the broadest pattern of a sentence in a natural language. Aris¬
totle’s syntax language specifies, abstractly, only nouns and verbs as its vocabu¬
lary, which are combined to form sentences according to this elementary rule. We
might wish that Aristotle had expressed this rule with at least the modest precision
here. However, Aristotle has neither a complete nor a complex set of syntax rules
of sentence formation in On Interpretation. Still, it is evident from his treatment
of this topic in his logical investigations that his understanding of the grammar of
a natural language is richer than his lack of rigorously stated rules would indicate.
And while this syntax rule is mixed with semantic notions, he nevertheless has
identified here the basic pattern of a sentence in a natural language.
Aristotle continues his discussion of grammar in On Interpretation 4 by focusing
his principal attention on the kinds of sentence that are subject to logical analysis.
He excludes, for example, prayers; and we take him also to exclude imperatives,
interrogatives, and exclamations (17a3-4). In Metaphysics 9.10, for example, he
makes this point rather emphatically: “for an affirmation and a sentence are not
the same [ou yap xauxo xaxdcpaau; xal cpdau;]” (1051b24-25). Accordingly, Aristo¬
tle considers only those kinds of sentence that are either true (dXrjfirjc; [ alethes ]) or
false (4»su5r)<; [pseudes]); or, as we express this nowadays, he considers only those
sentences that have a truth-value (16a9-13). His explicit interest is only with
the kind of sentence that expresses a proposition, namely, with the declarative
sentence. He writes:
18 Aristotle then drops this matter and remarks that it be treated at another place. He com¬
ments briefly on forming words in On lnt. 2-3.
Aristotle’s Underlying Logic
121
While every sentence [Xoyoc;] has meaning [arpavxixoc;], though not
by nature but, as we observed, by convention [auv0f]XT)v], not every
sentence is a declarative sentence [dxocpavxtxoc; Se ou tide;], but [only
those] to which being true or false belongs [dXX’ ev to dXr)0eueiv rj
(|teuSeCT0at undp/ct]. (16b33-17a3)
Aristotle uses ‘dxocpotvau;’ ( apophansis ) or ‘outocpavTixoc Xoyoc;’ ( apophantikos lo¬
gos) to denote the declarative sentence. A little later in his discussion he uses
‘xaxacpaau;’ ( kataphasis) and ‘ditocpaau;’ ( apophasis ) to denote two species of declar¬
ative sentence, namely, affirmation and denial, respectively. He uses ‘Xoyoc’ (lo¬
gos), and sometimes ‘cpdatc;’ ( phasis ), to denote the genus sentence, but he often
uses ‘ logos' (among its other uses) interchangeably with ‘apophansis’. Thus, while
a sentence consists in a noun and a verb, both of which are themselves meaningful
sounds or expressions established by convention and not by nature, 19 they do not
necessarily express something true or false. Truth and falsity involve predication
(16b7, 9-10), in particular for Aristotle, predicating one thing of another one thing
by combining (auvOeatc;) or dividing (Staipeaic;). Thus, with his discussion in On
Interpretation together with his fuller discussion of predication in Categories, Aris¬
totle names as genuine objects of logical investigation only those sentences that
involve predication so as to express a proposition. 20
Aristotle also recognizes a natural language to consist in both simple and com¬
pound sentences. Again, his logical investigation focuses on the simple sentence,
and in On Interpretation 5 he anticipates his treatment of sentences in Prior
Analytics.
One kind of declarative sentence [ditocpavau;] is simple [omXfj], that is,
it affirms or denies some one thing of another [xl xaxct xtvoc; f\ xi omo
xiv6<;], while the other is composite [auyxeipevr]], that is, a sentence
compounded [Xoyoc;... ctuv0exo<;] of [such] simple sentences. And [such]
a simple sentence is a meaningful expression, concerning something be¬
longing or not belonging [ecru 8’ rj pev dxXfj ditotpotvaic; cpcovf) arjpavxixf)
Tiepl too sl uitdpxet- xi i] pf) uxapyei.] in the different tenses. (17a20-24)
He states that “an affirmation and a denial are simple when they denote
[arjpouvouaa] some one thing of one other, whether or not universally or of some¬
thing universal [ptoc 8e eoxi xaxdcpaCTtc; xai drakpccaic; f) ev xa0’ evog arjpodvouaa, fj
xa0oXou ovxoc; xa0oXou r] [if) ojjoigx;] (18al2-13). Again:
19 That for Aristotle words and sentences have meaning by convention and not by nature see On
Int. 16a5-8, 16a26-28, and 16b33-17a2. Aristotle takes combining (auv0Ecu<;) to be conventional.
Moreover, since he was aware that Greek was one language among others, it seems likely that he
understood his metalinguistic posture toward language.
20 Aristotle in Cat. \ 2a4-10 complements what he writes here in On Int. “None of these
things mentioned [i.e., things said without combination] in itself is an affirmation; an affirmation
comes about in combination with other things. Every affirmation, it seems, is either true or false,
but of things said with no combination none is either true or false, for example, ‘man’, ‘white’,
‘runs’ or ‘wins’.” Cf. On Int. 10: 20a34.
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George Boger
An affirmation [xocxdcpaau;] is one that denotes [crjpouvouaa] something
of something. The subject [xouxo] is either a noun or a something not
having a name [an indefinite noun], and what is affirmed must be one
thing about one thing. (19b5-7)
Thus, in On Interpretation Aristotle recognizes two kinds of sentence (logos), in
particular, two species of declarative sentence, that express a proposition and that
are, accordingly, proper subjects of logical analysis (Figure 1).
Declarative sentence
orn:6<pavai<; or Xoyo<; onuxpavxixop [17a23 24]
xcfxdtpaau; [17a25] dmocpotme; [17a25-26; cf. 17b37-18al]
xaxacpaxixop Xoyoc; [12b6—7] axcxpaxtxop Xoyoc [12b8-9]
Figure 1.
Aristotle succinctly defines each of these in On Interpretation 6: “an affirmation
is a sentence affirming one thing of another; a denial a sentence denying one
thing of another [xocxotcpacm; §e eoxiv otxocpavatc xtvoc, axocpacnc Se saxiv axocpavaic
xivoc onto xivo;]” (17a25-26). In this connection we can extract a second sentence
formation rule for Aristotle, one pertaining especially to discursive discourse in
a natural language and, following Aristotle, we restrict this rule to the simple
sentence because it prepares us for his treatment of categorical sentences in Prior
Analytics.
SFR2 A simple declarative sentence in a given natural language is a sentence
that predicates one thing of another one thing, either attributively or
privatively, so as to have a truth-value.
Attributive (xaxr)yoptx6c) predication produces an affirmation, privative (axcp-
rjxixoc) predication a denial, and such a denial always involves a negative operator
( Pr. An. 51b31-35). We can represent the pattern of such a sentence graphically
as follows (Figure 2). 21
21 In On Int. 5: 17al3-15 Aristotle remarks that ‘two-footed land animal’ is one thing and
not many, but then refers this topic to another discussion. In any case, he here acknowledges
that a term (or non-logical constant) need not be a single word. He briefly takes up this topic
again in On Int. 11. There he claims that it is appropriate to combine predicates and subject
terms when the predicates are not accidental to the subject. His example is “for two-footed and
animal are contained in [Evcmapxsi Yap sv] man” (21al7-18). Aristotle always has his language
follow his theory of substance.
Aristotle’s Underlying Logic
123
Pattern of a generic natural language simple sentence
Simple noun
+
Simple verb (predicate)
Figure 2.
Aristotle notes in On Interpretation 10: 20bl-12 that the general word order in
natural Greek does not affect meaning: “nouns [subjects] and verbs [predicates] are
interchangeable [pexaxtGepeva] and express the same meaning [xctuxov crrjpdivet]”
(20bl-2). He provides some examples to establish that interchanging the place of
a noun and a verb does not generate two contradictions for a given sentence. He
concludes by reaffirming his meaning: “Thus, by interchanging the noun and the
verb an affirmation and a denial remain the same” (20bl0—11), or, that is, express
the same proposition. 22
In Sophistical Refutations Aristotle poignantly emphasizes this point about the
kind of predication specifically relating to the kind of discourse subject to logical
analysis. There he focuses on sentences used in argumentation particularly as
premisses. He writes in Sophistical Refutations 6, in connection with reducing all
fallacies to ignoratio elenchi:
And since deduction is based on [declarative] sentences [taken as pre¬
misses] [exd S’ 6 auXXoytapoc; ex xpoxaaeuv], and refutation [6 5’
e'Xeyxo?] is a deduction [auXXoyujpoc;], refutation will also be based
on [such] sentences [ex xpoxaaeov]. If, therefore, [such] a sentence [f)
xpoxaau;] is a single predication about a single thing [£v xa9’ evo?],
clearly this fallacy [viz., treating many questions as one] also depends
on ignorance of the nature of refutation; for what is not [such] a
sentence appears to be one [cpaivexai yap iivai xpoxaau; f] oux oOaa
xpoxaau;]. (169al2-16)
Of course, here Aristotle uses ‘xpoxaau;’ ( protasis ) to denote a sentence (logos,
apophansis) used as the starting point for argumentation, that is, to denote a
sentence in the role of a premiss. His discussion here comports exactly with his
practice in Prior Analytics and with his definition of ‘ protasis' there.
A premiss, then, is a sentence affirming or denying something about
something [npoxaau; pev oOv eaxi Xoyoc; xaxacpaxixoc; r] axocpaxixoc;
xivoe; xaxa xivoc;]. 23 ( Pr. An. Al: 24al6-17; cf. On Int. 11: 20b22-
25)
22 All this, of course, is easier for classical Greek than for, say, modern English, because Greek
is a highly inflected language.
23 We use R. Smith’s (1989) translation of Prior Analytics with some modifications, notably
translating ‘ sullogismos’ by ‘syllogism’ and not by ‘deduction’ in all cases.
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George Boger
He also notes in Prior Analytics A1 that “a syllogistic premiss without quali¬
fication will be either the affirmation or the denial of one thing about another
[earon auAAoyiaxixf) jaev Ttpoxaaig ootAdk; xaxatpamg t] ontocpaaig xivog xaxa xivog]”
(24a28-30). In Posterior Analytics A22 he writes that “one thing is predicated of
one thing [ev xa0’ evog xaxr]yopeLO0ai]” (83bl7-18; cf. Po.An. A2: 72a5-14).
While in On Interpretation Aristotle provides rules for sentence formation in
a natural language in a rather intuitive and, by modern standards, non-rigorous
manner, he nevertheless is especially concerned there with syntactic matters. He
even provides definitions of denial and affirmation that have a syntactic character,
although, again, they are mixed with semantic notions, and he cites examples to
bear out his meaning. He writes:
Whatever someone may affirm, it is possible as well to deny, and what¬
ever someone may deny, it is possible as well to affirm. Thus, it is ev¬
ident that each affirmative sentence has an opposite denial [outocpotaig
dtvxLxeipievifj], just as each denial has an [opposite] affirmative. (17a30-
33)
Here Aristotle is particularly concerned with contradictories and not with con¬
traries.
In this connection Aristotle provides in On Interpretation a rather syntactic
rule for forming the negation, or contradictory, of a given affirmation. This rule
is similar to that for forming an indefinite noun by prefixing ‘ou’ ( ou , ouk, oux )
to the common noun. 24 His example of negating a common noun is the following
(16a29-30):
avGpcottog (man) oux avOpcottog (non-man or not-man)
In On Interpretation 10: 20a7-9 he states that the 'ouk’ is attached to the noun
and is not a part of the verb: “but the ‘not’ must be added to ‘man’ [aAAa xo
ou, xfjv dmocpocaiv, xw avGpumog TtpocrOexeov]”. Table 1 provides two sets of his
examples of contradictories from On Interpretation 10, each having its one negation
(contradictory) matched with its one given affirmation (19bl4—19; cf. 19b27-29 &
19b38-20al).
It is interesting to notice that Aristotle does not cite a partial privative sentence
as the negation of a universal attributive sentence as is his customary practice when
he treats contradictories in Prior Analytics. Here he prefixes an entire sentence
with ‘ ou ’ in a syntactic way, and he does this both with the verb ‘to be’ and with
transitive and intransitive verbs. Thus, for “ndg eaxiv avOpwitog Slxaiog” (“Every
man is just”) in the above examples he could have used “Tig avBpwTtog oux ecm
24 In On Int. 2 (16a29-32; cf. On Int. 10: 19b7—10) Aristotle coined an expression for a noun
prefixed by ‘ou’ (oux, oux): ‘ovopa aopurrov. This is usually translated as ‘indefinite noun’ or
‘indeterminate noun’. He also named indeterminate verbs (On Int. 3: 16b 1 i—15) with a similar
prefix: ‘prjpa dopiaxov’.
Aristotle’s Underlying Logic
125
Table 1.
Affirmation
Negation
Egtiv avBpconoe-
[Man is.]
Ecttlv oux avBpwnoe-
[Man is not.]
Ecru note avGpcanoe.
[Every man is.]
Ecru note oux otvBpconoe-
[Every man is not.]
Oux ECTTLV otvBpconoe.
[Not - man is.]
Oux ecttlv oux dvBpconoe.
[Not - man is not.]
Oux ecttl nae avBpwnoe-
[Not - every man is.]
Oux ecttl note oux dvBpconoe-
[Not - every man is not.]
Also consider the followin
1 from 19b32-35 and 20a5-7:
Ilote ecttlv avBpojnoe §ixouo<;.
[Every man is just.]
Ilae scruv avBpwno e ou-SixaLoe-
[Every man is non-just.]
Tytcttvei note ctvGponoe-
[every man fares.]
Tytatvst nae oux-avBpconoe-
[Every non-man fares.]
Ou nae ecttlv avBpconoe SlxaLoe-
[Not - every man is just.]
Ou nae ecttlv avBpwnoe ou-SixaLoe-
[Not - every man is non-just.]
Ouy OyLaLVEL nae dvBpwnoe-
[Not - every man fares.]
Ouy OyLaLVEL nae oux-dvBpwnoe-
[Not - every non-man fares.]
SixaLoe” (“Some man is not just”) as its specific negation, but he did not. 25 In
this connection, then, we can formulate another rule of sentence formation having
to do with negation as follows (Table 1):
SFR3 The negation of a given affirmation in a natural language is formed by
prefixing ‘not’ to the entire sentence.
Prefixing l ou’ in this way to form the negation, or contradictory, of a given sentence
does not comport with ordinary Greek syntax, and thus it indicates an artifice on
Aristotle’s part in his treatment of sentences in On Interpretation. We can take
‘not’ here to mean ‘it is not the case that ... ’.
25 A strong test of Aristotle’s having conceived of negation in a syntactic way would be to find
instances of double negation, which we have yet to find. Another test would be to find instances
of his prefixing a sentence with ‘it is false that ... ’ or ‘it is not the case that ... Here there
seem to be ample cases. There is at least a kind of double negation of a term, i.e., of a simple
noun, in Po.An.All-. 77al7 18. Aristotle writes “pr| Cg>o v 8s p>i” and “pr) tjcoov 8’ ou”, which
J. Barnes translates as “not not an animal” (1994:17). Consider also On lnt. 9: 18bl 1—14.
126
George Boger
Although Aristotle only addressed sentence formation in a natural language in
On Interpretation, we can see him there having already anticipated the formal
language found in Prior Analytics.
Aristotle on proposition
Whether Aristotle had taken a philosophical position, vis-a-vis the modern discus¬
sion, on whether or not propositions exist as ideal objects need not concern us here.
Given his anti-platonic, materialist tendency this seems unlikely. However, if we
take ‘proposition’ more loosely to denote the meaning of a declarative sentence,
we can easily see that Aristotle made and worked with a distinction between a
sentence, which is a linguistic object, and the meaning or proposition it expresses,
which is a non-linguistic object. Both of these, of course, he distinguished from
what a sentence denotes, which is a state of affairs [ttpaypa], that is, something
that obtains or does not obtain in the world. This is evident from his treatment
of sentences in On Interpretation and in Prior Analytics, when he used ‘apophan-
sis’, and ‘ logos apophantikos’ , as well as ‘kataphasis’, ‘apophasis’, and ‘protasis’.
None of these words properly translates as ‘proposition’ per se; but each can be
understood to convey what ‘sentence expressing a proposition’ means. Aristotle
recognized that two or more different sentences, whether of one’s own natural lan¬
guage or of different natural languages, might express the same proposition, as
well as that the same sentence might express more than one proposition or have
more than one meaning. In On Interpretation 1 he writes that words are “symbols
of affections in the soul” (xuv cv if) cjtuxfj 7ta0r]pax«v cmppoXa), that speech, and
thus writing, is not the same for all peoples (16a3-6; cf. nl9 above). He continues:
But all the mental affections themselves [xauxa naox 7ta0r)paxa xfjc
cpuyf)*;], of which these words are primarily signs [ar)peia], are the same
for everyone, [just] as are the objects [rcpaypaxa] of which those affec¬
tions are likenesses [opoiupaxa]. (16a6-8)
Perhaps his claim here about the whole of mankind is a bit sweeping. Neverthe¬
less, he clearly indicates here his distinguishing very different linguistic objects as
expressing the same meaning or expressing the same proposition — that peculiar,
non-linguistic thing that is grasped by a human being in thought. 26
26 Aristotle makes an analogous remark in Po.An.A10 about demonstration: “Deductions, and
therefore demonstrations, are not addressed to external argument [ou yap xpoc; xov Xoyov]
but rather to argument in the soul [aXXa xpoc; xov ev xfj <puxji], since you can always object to
external argument, but not always to internal argument” (76b24—27). There is another similar
remark in Meta. 4-5 in connection with his arguing against those who disparage the law of non¬
contradiction: “For those who hold such opinions because they are confused by real difficulties,
can easily be cured of their ignorance by someone who addresses himself not to their arguments,
but to their meaning [ou yap jxpcx; xov Xoyov dXXa jipoq xf)v 5iavoiav]; whereas those who argue
for argument’s sake can be cured only by refuting each of their explicit arguments verbally by
other arguments” (1009al8-22). Also consider On the Soul 3.3-8 and SR 10 where Aristotle
treats the distinction some make between an argument in thought and an argument in words.
Aristotle’s Underlying Logic
127
Aristotle makes much the same point in On Interpretation 14, although there
in relation to considering what count as genuine contraries.
Is an affirmation [f] xaxacpaan;] contrary to a denial [xf omotpaaei]
or contrary to another affirmation [rj xaxacpaau; xfj xaxoccpaaei]? Is
the sentence [6 Xoyot;] “Every man is just” contrary to “No man is
just”? or to “Every man is unjust”? For example, “Callias is just”,
“Not - Callias is just”, “Callias is unjust”. Which of these sentences
are contraries?
For if the expression corresponds with things in the mind, and it is the
opinion of the contrary that is contrary, for example, that “Every man
is just” [is contrary to] “Every man is unjust”, then the same thing
must hold of our expressed affirmations as well [el yap xa psv ev xfj
cpwvf) axoXouGst xou; ev xfj Siavota, exet S’ evavxla Soifa f) xou cvavxlou
... xod etcI xwv ev xfj (pwvrj xaxacpaaewv avayxr) opolax; £)(£iv]. But if
it is not the case that the opinion of the contrary is not the contrary,
then neither will one affirmation be the contrary to another [el Se pr| 8 e
exel f) xou svavxlou Solja evavxla eoxlv, 0 O 8 ’ f| xaxacpaau; xf] xaxacpaaeL
Eaxai evavxla]; but the above mentioned denial will be the contrary
[aXX’ rj EiprjpEvr) arcocpaai;]. And so, one must inquire which opinion
is contrary to a false opinion, whether the opinion of the true denial
or the contrary opinion. [On taking ’good’ and ‘bad’ in sentences and
thought.] (23a27-23b2)
Aristotle here fusses with a notion of logical equivalence while considering different
sentences and their meanings. He takes up this matter in virtually the same way
in Prior Analytics A\6. In any case, this shows that he distinguished a sentence,
here taken as a formal object, from its meaning or content.
Aristotle specifically addresses the topic of a given sentence having more than
one meaning in On Interpretation 8. To establish a point of contrast he first
mentions some sentences in which each word has only one meaning (18al2-17). He
then cites an instance of a sentence in which a given word is artificially designated
as having more than one meaning. He writes:
If one word [ev ovopa] has two meanings, which do not combine to make
one, the affirmation itself is not one [oO pla xaxacpaai;]. If, for instance,
you give the name ‘garment’ alike to a horse and a man, then it follows
that “garment is white” would not be one but two affirmations, nor
would “garment is not white” be one denial but two. (18al8-21)
Surely this treatment of the topic not only indicates his making a clear distinction
between a sentence and the proposition it expresses, but it also strikingly rings of
his experimenting with a notion of reinterpretation familiar to modern logicians.
This is not to claim that Aristotle is a model-theoretic logician. Still, it is evident
that in this case Aristotle has retained the word ‘garment’ but reinterpreted it
twice. And while his example is elementary and serves as an illustration, he might
128
George Boger
just as easily have reinterpreted an entire sentence. In this connection it is worth
noticing that Aristotle regularly used the definite neuter article ‘to’ (inflected as
appropriate in the context of his exposition) as moderns use quotation marks to
mention a word or an expression or even to mention an entire sentence.
The matter of a given sentence expressing more than one proposition occupied
Aristotle’s attention in Sophistical Refutations, where his discussion is consider¬
ably more developed than has customarily been acknowledged. Surely recognizing
a given sentence as having more than one meaning is evidence of making a dis¬
tinction between its syntax and its semantics. Sophistical Refutations is replete
with such examples, especially, for instance, where Aristotle treated ambiguity and
equivocation. 27
We can now turn to Aristotle’s model of the grammar of the formal language
of the underlying logic depicted in Prior Analytics. His notation there is quite
elementary: he employed only upper case Greek letters as schematic placeholders
for terms in categorical sentences. And he never provided abbreviations for his
logical constants. Nevertheless, he specifically treated sentence patterns and their
logical relationships in a genuinely syntactic manner.
The grammar of Aristotle’s formal language
In respect of the theory of predicating of substance outlined in Categories, as
treated also in Metaphysics and Topics, and that underlies his notion of predication
in Prior Analytics, Aristotle understood there to be four ways that an attribute or
property —I'Siov, Ttd0o<;, xoiov (see Top. f.5.T02al8-30 on’t'Stov) — can be related
(or belong) to a substance or subject — ouata or Onoxdpevov (Table 2).
Table 2.
Kinds of substance attribution
1. Every individual of a given kind has a given property.
2. No individual of a given kind has a given property.
3. Some individual of a given kind has a given property.
4. Some individual of a given kind does not have a given property.
These attributions involve ontic relationships that exist independent of a knower:
they obtain or they do not obtain. Aristotle referred to such matters generally
27 In SR Aristotle treated the syllogistic deductive process as well, but there his focus was on
semantic matters. In particular, he treated the fallacies as though they formally violate what
a deduction ( sullogismos ) is as this topic is treated in Pr. An. For example, in the case of
ambiguity, while a given argument with an ambiguity has one grammatical pattern, which helps
to make it appear to be a deduction, it really has two underlying logical patterns. And in the
case of equivocation, while an argument with an equivocal expression has a given grammatical
pattern that makes it appear to be a deduction, it really has, with the addition of a fourth term
(in relation to a standard three term deduction), an underlying logical pattern different than a
deduction. Nothing results necessarily in these cases.
Aristotle’s Underlying Logic
129
as upaypanra ( pragmata ; singular pragma), or states of affairs, facts, and he used
‘to elvai’ — “to be [the case]” — and ‘to pirj eivoa’ — “not to be [the case]”
— to qualify them (cf. on his using ‘dtXrjGqc;’ and ‘t];eu§f|c;’ in this connection).
From these facts about existence Aristotle conceived four ways that a human
being could express — that is, predicate (xonnr)yop£t.v) — these substance/attribute
relationships linguistically. In the process he invented four logical constants to
capture these relationships — and he explicitly named each, although without
an expression for ‘logical constant’, in Prior Analytics A4 : 26b30-33 (cf. A23:
40b23-26). Thus, corresponding to the four ontic relationships above, there are
four possible predications of a subject by a participant (3).
Table 3.
Aristotle’s four logical constants
Logical Constant
Predication
Modern
Abbreviation
1. To navii UTtdpxsnv
(belongs to every)
A given property is pred¬
icated of [is said to be¬
long to] every member of
a given kind.
a
2. To prj&evi UTtapyeiv
(belongs to no)
A given property is pred¬
icated of [is said to be¬
long to] no member of a
given kind.
e
3. To tLvi UTtapysi-v
(belongs to some)
A given property is pred¬
icate of [is said to be¬
long to] some member of
a given kind.
i
4. To pf) xtvi uitapxEiv
(does not belong to
some)
To (af) Ttavxi
OitapxELv (belongs
not to every)
A given property is not
predicated of [is said not
to belong to] some mem¬
ber of a given kind.
o
Correspondingly, there are four categorical sentence patterns that Aristotle used
throughout his logical investigations in Prior Analytics. His most commonly used
schematic representations of the four categorical sentences are represented in Ta¬
ble 4.
Concerning any categorical sentence AB, then, A can be taken, or predicated,
of B in four ways. ‘A’ and ‘B’ here are schematic letters that hold places for
terms, or non-logical constants. The four kinds of sentence involve the four kinds
130
George Boger
Table 4.
Aristotle’s model for each kind of categorical sentence
Sentence pattern
Kind of categorial
sentence
Modern expression
1. to A xavTi t£> B
UTtdpxet (A belongs to
every B)
Universal attributive
AaB
2. to A prjhev'i tw B
UTtaxet (A belongs to
no B)
Universal privative
AeB
3. to A Tivi tw B uTtdxei
(A belongs to some
B)
Partial attributive
AiB
4. to A Ttv't tw B pi)
Okc(X £L (A does not
belong to some B)
Partial privative
AoB
of predication, which themselves reflect the four ontic relationships. 28 Aristotle
thought of a categorical sentence as having a special pattern that distinguishes it
from other kinds of sentence, namely, from those of natural Greek. Moreover, he
thought of each of the four kinds of categorical sentence as itself fitting a special
pattern. This is most evident in his treating the syllogisms and non-syllogisms in
Prior Analytics A4~6. Thus, we can extract Aristotle’s syntax rule, according to
his formal grammar, for forming a simple categorical sentence in a given object
language pertaining to a given domain and express it as follows.
CSFR1 A categorical sentence in a given language consists in combining a non-
logical constant with any one of the four logical constants with another
non-logical constant in this order.
This rule identifies the pattern of a categorical sentence in Aristotle’s formal lan¬
guage. Aristotle’s expression for ‘non-logical constant’ is ‘opot;’, or ‘term’. The
term in the first position is called the predicate term (to xaxrjYopoupevov or opoc,
xcrcr)Yopou[i£vov), the term in the second position is called the subject term (to
UTtoxeipevov or opo<; uxoxeijievoc;). In natural Greek it is customary, but not a
strict practice (On Int. 20bl—12), to place the subject of a sentence before the
predicate/verb. But in Prior Analytics we see that Aristotle quite deliberately
placed the predicate term before the logical constant, which acts as a verb, and
28 Aristotle takes the following expressions to amount to the same thing: ‘A belongs to every
B’ and ‘A holds of every B’; ‘A follows all B’; and ‘A is predicated of every B’. While Aristotle
used ‘umpXEav’ (huparchein) , ‘&xoXou9eiv’ and ‘xcrnrjYopEiaGai.’ respectively, his preference to use
1 huparchein 1 for the logical constants was not accidental but an important reflex of his theory of
substance.
Aristotle’s Underlying Logic
131
then place the subject term after the logical constant. In addition, the logical con¬
stants themselves are rather artificial constructions aimed to reproduce linguisti¬
cally what he took to be conditions of being, as in Categories and Metaphysics. It is
evident that Aristotle thought of a categorical sentence as formally constructed by
concatenating, stringing or combining, a predicate term (or non-logical constant)
with a logical constant with a subject term (or non-logical constant) strictly in
this order.
We might also extract two additional categorical sentence formation rules that
have a rather more semantic character, but which nevertheless bear on the logical
pattern of a sentence.
CSFR2 The two non-logical constants in every categorical sentence are not iden¬
tical. 29
CSFR3 The two non-logical constants in every categorical sentence are homo¬
geneous with respect to grammatical category, that is, both non-logical
constants are substantives.
This seems to be Aristotle’s practice, at least, for the most part. There is a passage
in Prior Analytics A36 that confirms this (CSFR3).
For we state this without qualification about them all: that terms
must always be put in accordance with the cases of the nouns [xcrra
xac; xXfjcreic; xov ovopaxcov] ... (48b40-41; see 48b39-49a2; cf. Pr. An.
A39-40).
Conversion otherwise would seem unintelligible. According to modern standards,
we might also formulate a fourth rule, which is surely implicit in Aristotle’s think¬
ing.
CSFR4 Nothing is a categorical sentence except in virtue of these rules.
29 This is a restriction on the language that anticipates its use in scientific discourse. However,
Aristotle does recognise identity in his formal logic per se (see, e.g. Pr. An. A 4 I and B15). There
is an interesting instance of Aristotle’s treating this matter in Po. An. A3 where he establishes
the inadequacy of circular reasoning for demonstration (but not its invalidity). He writes, after
remarking that he can accomplish this demonstration by using three or even two terms: “When
if A is the case, of necessity B is, and if B then C, then if A is the case C will be the case. Thus,
given that if A is the case it is necessary that B is the case and if B is the case that A is the
case (this is what is to proceed in a circle [however long the loop]), let A be C. Hence, to say
that if B is the case A is the case is to say that C is the case; and to say this is to say that if
A is the case C is the case. But C is the same as A. Hence, it follows that those who declare
that demonstrations may proceed in a circle say nothing more than that if A is the case A is the
case. And it is easy to prove everything in this way” (72b37-73a6). Surely Aristotle considered
such sentences as “Every horse is a horse”, or, metalogically, “Every A is an A”. However, since
he was preeminently concerned with ‘deriving something other’ (which requirement appears in
his definition of ‘ sullogismos’) for the proposes of extending knowledge in the sciences, such a
restriction on the language makes perfect sense and does not do violence to his logical acumen.
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There is no analogous, strong syntax rule for forming the negation, or contradic¬
tory, of a given affirmative sentence in Prior Analytics as there is at places in On
Interpretation. However, in Prior Analytics A46 he holds that denials require the
use of a negative operator.
Consequently, it is evident that ‘is not-good’ is not the denial [arcocpotan;]
of ‘is good’. If, therefore, ‘affirmation’ or ‘denial’ [f] cpdai<; f] &KOcpo«CT(.<;]
is true about every single <predicate>, then if ‘is not-good’ is not a de¬
nial, it is evident that it must be a sort of affirmation [xotxacpamc;]. But
there is a denial of every affirmation [xaracpaaeuc; Se itaar)c; axocpaaK;
ectilv], and, therefore, the denial of this affirmation is ‘is not not-good’.
(51b31-35).
For Aristotle a genuine denial, as distinguished from an affirmation, involves a
negative operator, whether as an adverb attached to a verb (predicate), or as a
pronominal adjective attached to a non-logical constant (or as part of the logical
constant). 30
We can represent Aristotle’s thinking on sentence formation as prescribed in his
formal language as follows (Figure 3):
Generic categorical sentence pattern
non-logical con¬
stant: predicate
1
non-logical con¬
stant: subject
+
logical constant
1 +
term
term
Figure 3.
Now, in Prior Analytics, as contrasted with his treatment of sentence formation in
On Interpretation, Aristotle fixed the word order in a sentence. The order of the
constituent parts of a categorical sentence does not change as it might in a natural
language. The syntax of a categorical sentence is strict since its use is anticipated
in the syllogistic deduction process, and this process requires precision.
In connection with treating language formally, Aristotle frequently writes of
‘taking’ (Xappdveiv) or ‘not taking’ A of B in one of four ways. He writes, for
example, about predicating in general that it is necessary to take something of
something: dvdyxr) Xapav xl xaxa xlvoc (Pr. An. A23: 40b31). He often uses‘AB’,
or similar expressions, to indicate any categorical sentence (Pr. An. ASS: 42b6).
This way of addressing predication tends to treat a sentence as uninterpreted,
although, of course for Aristotle, not completely. A categorical sentence may be
30 See below this section on Aristotle’s distinguishing an opposite sentence from an affirmation
and a denial. In short, while contraries are opposites (as are contradictories), each contrary
sentence might be an affirmation, that is, designating attribution and not privation, as in “Every
man is good” and “Every man is bad”.
Aristotle’s Underlying Logic
133
understood to express taking one term about another term as a formal matter.
This is especially the case in Prior Analytics A23 , which is an especially proof-
theoretic chapter (§5.1). Moreover, he often writes in the same manner about
taking sentences of one or another pattern, for example, as starting points of
argumentation. In addition, he often uses the word ‘xpopXrjpoi’ ( problema ) to
indicate, not a particular sentence with a particular meaning, but to refer to each
of the four kinds of categorical sentence. Consider in Prior Analytics A\ where
he uses ‘ problema ’ to indicate a sentence pattern:
All the problemata are proved through this figure [xai oxi ndvroi TtpopXrjfioaa
Seixvutou 81 a xouxou too axrjpotxoc;]. (26b30-31; cf. A27: 43al6-19,
A28: 44a36-37, k A29: 45a36-38)
‘ Problemata ’ here does not refer to problems in a given domain as he uses this
word in, for example, Problems and elsewhere, but to the four kinds of categorical
sentence: it is used purely in reference to a formal object. We see that Aristotle’s
formal language, at least in respect of sentences, while not a purely uninterpreted
object (as in string theory), is nevertheless sufficiently formal to exemplify the
defining structures or patterns of categorical sentences . 31 This indicates that Aris¬
totle took his logical (formal or artificial) language represented in Prior Analytics
to be a syntactic object for the purpose of defining an underlying logic.
Aristotle also considered the relationship of negation in a somewhat syntactic
manner, notwithstanding that his semantics is just below the surface. He writes
in On Interpretation that whatever can be affirmed can also be denied, whatever
denied can be affirmed, and that each attributive sentence and each privative sen¬
tence has its own opposite (17a30-33). This is all formal. In On Interpretation 1
he recognized two kinds of sentence that are opposites (tot dvxi.XEi.pEva): contradic¬
tories (od dvxupdasu;) and contraries (tcc Evavxta). This corresponds exactly with
what he writes in Metaphysics 5.10 on contrariety in things. In On Interpretation
7 he defines contradictories in a loosely syntactic manner as follows:
Now, I call an affirmation [xaxdcpacriv] contradictorily opposed [dvxicpaxLxdic;
avxixeiaGod.] to a denial [dnocpacnv] when what the one denotes [arjpdtvouaav]
universally and the other not universally. (17bl6—18)
And then he provides some examples
“Every man is white” [to] “Not every man is white” [ndc; avGpoTioc;
Xeuxoc — ou Tide; dv 0 pco 7 to<; Xeuxoc;] and “No man is white” [to] “Some
31 In addition, Aristotle uses three sets of schematic letters to mark places for terms, one set
for each figure: for the first ABF (ABC), the second MNE (MNX), and the third 77 RE (PRS).
He names terms by their schematic positions — first (or major), middle, last (or minor) — and he
calls the first and last axpa (extremes). His use of the terms ‘axi)pa’ (figure or arrangement) and
‘Siaaxripa’ (interval) are further indications. In addition, in his practice of substituting actual
terms for schematic letters when determining that a premiss-pair pattern is inconcludent, he sets
out such terms according to the schematic order for each figure’s schematic letters: first figure
— PMS; second figure — MPS; third figure — PSM.
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man is white” [ouSdc; avfipwiioc; Xeuxoc — ecm tic; ocvOpoKoc; Xeuxot;].
(17bl8-20)
This definition is used throughout Prior Analytics. Aristotle models these sen¬
tences as shown in Table 5.
Table 5.
Contradictories
Aristotle’s text
Modern notation
1. A belongs to every B— A does not belong to some B.
2. A belongs to no B — A belongs to some B.
AaB —o AoB
AeB — AiB
The following two syntactic relationships hold between these different sentences:
1. Whenever a sentence fitting the pattern AaB is taken, then a sentence fit¬
ting the pattern AoB cannot be taken; and whenever a sentence fitting the
pattern AoB is taken, then a sentence fitting the pattern AoB cannot be
taken.
2. Whenever a sentence fitting the pattern AeB is taken, then a sentence fitting
the pattern AiB cannot be taken; and whenever a sentence fitting the pattern
AiB is taken, then a sentence fitting the pattern AeB cannot be taken.
Here Aristotle leaves the schematic letters uninterpreted — or unsubstituted —
and asserts the formal, logical relationships that exist between sentences fitting
such patterns. In fact, from his text on contradictories (cited above), we can
extract Aristotle’s rule for their formation and express it as follows.
Contradictory formation rule
The contradictory of a given sentence, whether attributive or privative, is
formed by retaining the predicate and subject terms (non-logical constants)
as given and replacing the logical constants as follows:
1. In the case of a universal attributive sentence, the universal attributive log¬
ical constant is replaced by the partial privative logical constant.
2. In the case of a universal privative sentence, the universal privative logical
constant is replaced by the partial attributive logical constant.
3. In the case of a partial attributive sentence, the partial attributive logical
constant is replaced by the universal privative logical constant.
Aristotle’s Underlying Logic
135
4. In the case of a partial privative sentence, the partial privative logical con¬
stant is replaced by the universal attributive logical constant.
In On Interpretation 7 Aristotle defines contraries in a loosely syntactic manner
in the following way:
I call a universal affirmation and a universal denial contrarily op¬
posed [evavTttoc; (dvcixetafioti) he xrjv xou xaGoXou xaxctcpaaiv xal xr)v
too xaGoXou aKOcpacuv] (17b20-21)
His example is the following:
“Every man is just” [to] “No man is just” [itac avGpcmoc; blxctux; —
oOSe'u; avOpcmoc Slxouoc;]. (17b21—22)
Of course, in both contradictory sentences and contrary sentences, the subject
terms and the predicate terms in the one are the same in the other. Aristotle
indicated this a little earlier in On Interpretation 7:
Now if someone states universally of a universal that something belongs
or does not belong [oxi uxapxet. j) urj], there will be contraries. (17b3-5)
Perhaps he states this more emphatically in On Interpretation 14 : “for contraries
are among things that differ most in respect of the same thing [itepl to auxo]”
(23b22-23); and again in Categories 11 : “the nature of contraries is to belong to
the same thing, either in species or in genus [rap! tccjtov 7 ] ei&ei r] yevei]” (14al5-
16). In On Interpretation 6: 17a33-37 (34-35) he writes on contradictories that:
“I mean opposites [dvxiXEiaGou] that [affirm and deny] the same thing of the same
thing [xpv xoG auxou xaxa xou auxou] and not ambiguously [pf) opwvupwt;]”. This
definition is also used throughout Prior Analytics. Aristotle models these sentences
as in Table 6.
Table 6.
Contraries
Aristotle’s text
Modern notation
A belongs to every B — A belongs to no B.
AaB — AeB
The following syntactic relationships hold between the different sentences:
1. Whenever a sentence fitting the pattern AaB is taken, then a sentence fitting
the pattern AeB cannot be taken.
2. And whenever a sentence fitting the pattern AeB is taken, then a sentence
fitting the pattern AaB cannot be taken.
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We can extract a formation rule for contraries analogous to that for contradictories
from Aristotle’s text (cited above) and express it as follows.
Contrary formation rule:
The contrary of a given sentence, whether attributive or privative, is formed
by retaining the predicate and subject terms as given and replacing the
universal logical constants as follows:
1. In the case of a universal attributive sentence by replacing the universal at¬
tributive logical constant with the universal privative logical constant.
2. In the case of a universal privative sentence by replacing the universal pri¬
vative logical constant with the universal attributive logical constant.
What Aristotle writes in On Interpretation and in Categories corresponds exactly
with what he does and with what he writes in Prior Analytics B15 about the
formal relationships among categorical sentences.
I say that verbally [xaxa xqv Xe£iv] there are four (pairs of) oppo¬
site sentences [itpoxdaeu;], to wit: ‘to every’ [Travel] (is opposed) to
‘to no’ [o08evt]; and ‘to every’ [navel] (is opposed) to ‘not to every’
[ou Tcavxl]; and ‘to some’ [xivl] (is opposed) to ‘to no’ [ou5evl]; and ‘to
some’ [xlvI] (is opposed) to ‘not to some’ [ou xivl]. In truth, however,
there are three, for ‘to some’ and ‘not to some’ are only opposites ver¬
bally. Of these, I call the universal sentences contraries (‘to every’ is
contrary to ‘to none’, as, for example, ‘every science is good’ [xacrav
eTuafjprjv £tvai anou§aiav] is contrary to ‘no science is good’ [pr^Epiav
Sivai anouScaav]) and the other pairs of sentences opposites [sc., con¬
tradictories]. (63b23-30)
Aristotle’s concern for argumentational skill and logical syntax
The syntactic character of Aristotle’s treatment of opposition is all the more as¬
sured when we place his logical investigations in the context of his concern with
argumentation, as it pertains to both axiomatic discourse and disputational dis¬
course. Aristotle was eminently occupied in Sophistical Refutations and Topics
with equipping his students with argumentational skills that they could employ
quickly and with facility and keenness. These two treatises surely served as student
handbooks. Perhaps his introduction to Topics exemplifies this concern.
The purpose of the present treatise is to discover a method [pcBoSog]
by which we shall be able to reason deductively [aukkoylCcaBai] from
generally accepted opinions about any problem set before us and shall
Aristotle’s Underlying Logic
137
ourselves, when sustaining an argumentation, avoid saying anything
self-contradictory [uitevavxiov]. 32 (100al8-21; cf. Top.l. 2)
Aristotle did not have his students memorize certain texts — a set of specific, stock
speeches — to acquire this skill, as was a common practice at the time. Rather,
he expected them to become familiar with the structural — formal or syntactic —
aspects of cogent and fallacious reasoning. In effect, Aristotle aimed to have his
students become accomplished logicians. His formal interests are especially evident
in his closing remarks in Sophistical Refutations 3f. He writes, in connection with
remarking that his logical investigations are entirely new:
For the training given by the paid teachers of eristic argumentation
resembled the pedagogy of Gorgias. For some of them required their
students to learn by heart speeches that were either rhetorical or con¬
sisted of questions and answers, in which both sides thought that the
rival argumentations were for the most part included. Hence the teach¬
ing that they gave to their students was rapid but unscientific [ccxex v oc;];
for they conceived that they could train their students by imparting to
them not an art but the results of an art ... he has helped to supply
his need but has not imparted an art [xexvt)v] to him. ... [While there
was much information available having to do with rhetoric] whereas
regarding deductive reasoning [too auXXoYlCcaQat] we had absolutely
no earlier work to quote but were for a long time laboring at tentative
researches. (183b36-184b3) 33
Aristotle also took up developing argumentational skills in Prior Analytics , es¬
pecially at A24-46, the chapters that follow the formal representation of his de¬
duction system. He was particularly concerned in these chapters with developing
an individual’s ability to establish (xaxaaxEudCetv) or to destroy (dvaaxeuaCEtv)
an argumentation. 34 This theme is wholly consonant with his treatment of ar¬
gumentation in Sophistical Refutations and Topics. Indeed, the title of his works
on formal logic, ‘xd dvaXuxtxa’ — a topic specially treated in Prior Analytics
A45 — signifies his concern with the formal aspects of argumentation. Analysis
(dvdXuCTu;; dvaXustv) is a process of transforming one syllogism in any one figure
into another syllogism of another figure if both syllogisms prove the same problema
32 We use E. S. Forster’s (1960) translation of Topics and below his (1955) translation of So¬
phistical Refutations with significant modifications. Cf. L.-A. Dorion’s (1995) French translation
of Sophistical Refutations.
33 This passage continues and ends the treatise with the following remark that might have been
addressed to modern critical readers. “If, therefore, on consideration, it appears to you that, in
view of such original conditions, our system is adequate when compared with the other methods
which have been built up in the course of tradition, then the only thing which would remain for
all of you, or those who follow our instruction, is that you should pardon the lack of completeness
of our system and be heartily grateful for our discoveries” (184b3—8).
34 See Pr. An. A26-28 and summary at A30-. 46a3-10. For example, Aristotle writes ( A26 ):
“... a universal positive problema is most difficult to establish [xaTaaxeudaou] but easiest to
refute [dvaaxeudaoo.]” (43al-2).
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(§6.2). Aristotle aimed to promote his students’ facility with reasoning syllogis-
tically to establish and to refute arguments by studying the logical relationships
among sentence patterns and among patterns of elementary arguments. This is
analogous to a modern logician’s studying the formal relationships among the rules
of propositional logic.
Aristotle’s mathematical disposition toward the study of grammar
In On Interpretation Aristotle treated sentences in natural languages metalinguis-
tically. His practice there is much the same, although without the complexity,
as that of a modern grammarian whose natural language is, say, English, and
who writes an English grammar. Aristotle used Greek to mention Greek as this
grammarian would use English to mention English. However, Aristotle went con¬
siderably farther than a grammarian in his treating the syntactic aspects of a
language because he thought of his linguistic investigations as laying an epistemo¬
logical — or formal — foundation for various axiomatic sciences, the apodeiktikai
epistemai. As a logician formalizing a deduction system, Aristotle continued in
Prior Analytics to develop a formal grammar where a grammarian of a natural
language might leave off in On Interpretation.
Aristotle always took Greek, whether explicitly or implicitly, as the background
language for discourse in any specialized domain. In this connection, we might say
that Aristotle took Greek as his master language, although he never formulated the
matter using just such an expression. Still, he recognized that each of the special
axiomatic sciences was equipped, or ought to be equipped, with its own specialized
terminology, or vocabulary, appropriate to its domain. Aristotle indicated his
having a notion of a specialized vocabulary in Metaphysics f.2. There he wrote
about using terminology across sciences and thus indicated, albeit negatively, that
each science has its own terminology.
For a term belongs to different sciences, not merely because it is used in
many ways, but when its definition can be referred neither to a single
subject matter nor to a common ground. (1004a24-25)
That Aristotle had a clear notion of universe of discourse, although, again, without
an equivalent expression in Greek, is evident from his treatment of the genus of a
given science in Posterior Analytics A 7, 9-10, 28. 3 ° This notion is poignantly ex¬
pressed also in Metaphysics 10.4 '■ “ a single science covers a single genus and there¬
fore deals with the complete differences in that genus” (1055a31-32). 36 Perhaps
Posterior Analytics A1 (cf. A8-10 ) expresses his notion of universe of discourse
most plainly.
35 Cf. Aristotle’s treatment of different discourses in the various branches of mathematics in
Po. An. A5.
36 Cf. what Aristotle writes in Meta. J.2: 1003bl2-15 when considering the subject of phi¬
losophy and being qua being: “so whatever is said of one subject matter belongs to one science.
Accordingly, whatever is said in reference to a single nature is a single science; for such state¬
ments, too, in some way or other, refer to a single subject matter”. Cf. Pr. An. A30: 46al5-22.
Aristotle’s Underlying Logic
139
Thus you cannot prove anything by crossing from another kind —
for example, something geometrical by arithmetic. There are three
things involved in demonstrations: one, what is being demonstrated,
or the conclusion [to aupTtepaapa] (this is what holds of some kind in
itself); one, the axioms [rot odpcopaxa] (axioms are the items from which
the demonstrations proceed); third, the underlying kind [to yevoc; to
Cmoxetpevov] whose attributes [xa Tid0r)] — that is, the items incidental
to it in itself — the demonstrations make plain.
Now the items from which the demonstrations proceed may be the
same [xa aCrca]; but where the kinds are different [to yevoc; exepov], as
with arithmetic and geometry, you cannot attach arithmetical demon¬
strations to what is incidental to magnitudes — unless magnitudes
are numbers. ... Arithmetical demonstrations always contain the kind
with which the demonstrations are concerned, and so too do all other
demonstrations. Hence the kind must be the same, either simpliciter
or in some respect, if a demonstration is to cross [coax’ f) omXax; avayxr]
to auxo £tvat yevop f) itrj, et peXXet fj dmoSa^tp pexapaiveiv]. That it is
impossible otherwise is plain; for the extremes and the middle terms
must come from the same kind [ex yap too auxou yevoup avayxr) ia
axpa xat xa peaa £vat], since if they do not hold in themselves, they
will be incidentals. 37
For this reason you cannot prove by geometry that there is a single sci¬
ence of contraries, nor even that two cubes make a cube. (Nor can you
prove by any other science what pertains to a different science, except
when they are so related to one another that the one falls under the
other — as, for example, optics is related to geometry and harmonics
to arithmetic.) Nor indeed anything that holds of lines not as lines
and as depending on the principles proper to them — for example,
whether straight lines are the most beautiful of lines, or whether they
are contrarily related to curved lines; for these things hold of lines
not in virtue of their proper kind but rather in virtue of something
common. (75a38-75b20)
Not only has Aristotle indicated his notion of universe of discourse in relation to a
genus, but he has also indicated that he worked with a notion of category mistake.
This matter also is treated in Sophistical Refutations.
That different scientific domains are distinguished in one or another discourse
is an important part of Aristotle’s discussion of fallacious reasoning in Sophisti¬
cal Refutations. There he treats a kind of fallacious reasoning that violates the
boundaries of different domains. In Sophistical Refutations 8: 169b20-23 he re¬
marks that a sophistical refutation, while it is usually a spurious deduction of the
37 In Po. An. AS: 74b27-33, for example, Aristotle made this point about the different genera
in an interesting way by stating that when the middle term of a deduction is necessary the
conclusion must be necessary and thus germane to one science and not to another. ‘Necessary’
here is used in its modal sense.
140
George Boger
contradictory of a given sentence, might, nevertheless, be a genuine deduction (i.e.,
a refutation) but one that is not germane to the subject matter under discussion.
The deductive reasoning while genuine “only seems to be, but is really not, ger¬
mane to the subject at hand [aXXa xod xov ovxa piev cpaivoptevov §e olxelov xou
TtpdYjxaxoc]”. In Sophistical Refutations 9 Aristotle draws a distinction between
the function of a scientist and that of a dialectician, or, that is, of a logician. In
this connection he writes about demonstrations and refutations special to a given
science.
So we shall need to have scientific knowledge of everything; for some
refutations will depend on the principles of geometry and their con¬
clusions, others on those of medicine, and others on those of the other
sciences. Moreover, spurious refutations [ot ([jeuSe'.c; eXeyxol] also are
among things which are infinite [as are, perhaps, the sciences and their
demonstrations (170a22—23)]; for every art has a spurious proof pecu¬
liar to it, geometry a geometrical proof and medicine a medical proof.
By ‘peculiar to an art’ [xo xaxot xf)v xexvqv] I mean ‘in accordance with
the principles of that art’ [xo xaxot xac exeLvr)<; apyap]. (170a27-34)
Aristotle returns to this matter in force in Sophistical Refutations 11, where he
treats the discipline of logic as the dialectical art of the deductive principles com¬
mon to all intelligible, cogent discourse. He also distinguishes the sophist from the
eristic in respect of their motives (171b29-34). In this connection he establishes a
clear notion of universe of discourse.
Then there are those spurious deductions that do not accord with the
method of inquiry peculiar to the subject yet seem to accord with the
art concerned. For false geometrical figures are not contentious (for the
resultant fallacies accord with the subject-matter of the art), and the
same is the case with any figure illustrating something which is true, for
example, Hippocrates’ figure or the squaring of the circle by means of
lunules. ... [Bryson’s method of squaring is sophistical because it does
not accord with the subject-matter.] ... And so any merely apparent
deduction on these topics is a contentious argumentation, and any
deduction that merely appears to accord with the subject-matter [xaxct
xo TtpaYpa], even though it be a genuine deduction, is a contentious
argumentation [because it only appears to accord with the subject-
matter]. (171bl1—22)
A little later in this discussion Aristotle provides some examples to illustrate his
meaning.
For example, the squaring of the circle by means of lunules is not con¬
tentious, whereas Bryson’s method is contentious. It is not proper to
transfer the former outside the sphere of geometry because it is based on
principles that are special to geometry [xou xov ptcv oux ectxl piExevEYxexv
Aristotle’s Underlying Logic
141
aXX’ q jtpoc YEopexplav povov, Sta to ex x«v iStcov Eivou apycov], whereas
the latter can be used against many disputants, namely, all those who
do not know what is possible and what impossible in any particular
case; for it will always be applicable. And the same is true of the way
in which Antiphon used to square the circle. Or, again, if someone
were to deny that it is better to take a walk after dinner because of
Zeno’s argumentation, it would not be a medical argument; for it is of
a general application. (172a2-9)
It is evident, then, that Aristotle recognized there to be any number of special
sciences, each with its own domain and topical sub-language, each of which is a
fragment of a whole, or master, language.
Aristotle’s focus shifts from a general concern with grammar, as in On Inter¬
pretation , to a more specialized concern with language and grammar in Prior
Analytics and Posterior Analytics and in Metaphysics. There language is treated
(1) as modified from natural language according as its universe of discourse is
delimited and specialized and (2) as more rigorously formalized for the purposes
of precision and deduction. Aristotle’s emphasis on the simple sentence in On
Interpretation repays him well in Prior Analytics where he treats his deduction
system with more rigor. His linguistic and argumentational analyses in On Inter¬
pretation, and in Sophistical Refutations and Topics , provided the foundation for
his formulating the simple grammar of the artificial language in Prior Analytics.
Now, while Aristotle seemed always to have natural language in the background
when he undertook his logical investigations, his thinking was surely disposed to¬
ward constructing an artificial language. And while he surely did not work with a
fully uninterpreted calculus, he nevertheless had already moved toward developing
a notion of a precise scientific language for extended deductive discourse in each
of the special axiomatic sciences. And here he developed some stringent require¬
ments for intelligible discourse. In particular, in Metaphysics 11.5 he expressed a
requirement that in scientific discourse one word have one meaning, and if it were
to have more than one meaning this should be made patently clear. He treats
this topic there in conjunction with treating the law of non-contradiction in the
following way:
Those, therefore, who are to communicate with one another by way
of argumentation [Xoyou] must have some common understanding.
... Each word must therefore be intelligible [yvwpipov ev] and indi¬
cate something definite, not many things, but only one [xod ^tf) TtoXXd,
povov 8c ev]; and if it has more than one meaning it must be made
plain in which of these the word is being used. He, therefore, who says
that “this is and is not” denies what he affirms, with the consequence
that he declares the word to signify what it does not signify; but this
is absurd. Consequently, if “this is” signifies something, it is absurd to
assert truly its contradictory.
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Accordingly, if a word signifies something, and this is truly asserted,
this [connection] must be necessary; but what necessarily is cannot ever
not be; and so opposite sentences concerning the same thing cannot be
true together ... [again] opposite sentences concerning the same thing
can never be true together. (1062all-23) 38
Aristotle’s scientific languages eschew the ambiguity that abounds in natural lan¬
guages. Surely his concern for a precise syntax is a reflex of his concern for scientific
precision. Again, we have the testimony of Sophistical Refutations to make Aris¬
totle’s case; this point is especially evident there when Aristotle treats fallacious
reasoning involving ambiguity and equivocation as well as making many questions
into one.
In connection with his study of logic, then, what he writes in Metaphysics and
in Sophistical Refutations can be taken as a move toward developing a logically
perfect language , albeit restricted in its scope to a specific, delimited domain. Nev¬
ertheless, Aristotle’s impulse and that of a modern logician are correspondingly
the same when their respective focuses are on the deduction process. Now, of
course, each of these scientific languages is a topical sub-language of a given mas¬
ter language, in this case Greek, and as such each is an object, if not a natural,
language. We might think of this as natural Greek departmentalized; or, rather,
that the languages of mathematics and biology, for example, are specialized topi¬
cal sub-languages with a tailored Greek as their mode of expression. In any case,
Aristotle’s requirements for scientific discourse, in connection with his syllogistic
logic, indicate that his treatment of language in On Interpretation, Prior Analyt¬
ics, Metaphysics, Sophistical Refutations, and even Categories is (1) thoroughly
metalinguistic and, thus, (2) especially occupied with syntax. Aristotle aimed to
formalize scientific discourse, not only with a polished deduction system, but also
with precisely formulated linguistic requirements concerning both its syntactic and
semantic dimensions.
Aristotle’s formal language is not strictly an uninterpreted calculus awaiting
an interpretation as a modern logician understands this matter. Its vocabulary
consists only in (1) four fully interpreted logical constants and (2) a number of
schematic (upper case Greek) letters that function purely as metalinguistic place¬
holders for terms in categorical (or predicational) sentences. These schematic let¬
ters, however, are generally uninterpreted in a way familiar to modern logicians;
but they are not variables. 39 There are no genuine variables, whether bound or
free, ranging over individuals in a given domain in Aristotle’s formal language.
Indeed, there is no need for variables, since the system lacks quantification theory
and works with patterns appropriate to a term logic. Nor, then, are there any non-
38 See Cat. 1 on equivocal (on<ovuiro<;) and univocal (auvuvu[iot;) names; cf. Cat. 5: 3b7-9.
Also cf. Meta. 4-4'■ 1006a28-1006b20 in the context of defining ‘having a meaning’.
39 J. Corcoran (1974: 100) has called these “metalinguistic variables”; cf. R. Smith (1984: 590,
595) who refers to them as “syntactic variables for terms”. We believe that Aristotle takes his
letters to be schematic letters in a way similar to W. O. Quine’s meaning of “dummy” letters
(1970:12; 1982: 33, 145-146, 160-162, 289, 300-301).
Aristotle’s Underlying Logic
143
logical constants in his formal language. Non-logical constants pertain to a given
universe of discourse along with its object language. Again, there is no need for
any non-logical constants. His formal language does not anticipate quantification
and the existence of variables in a given object language.
It may be that considerations of natural language underlay Aristotle’s thinking
when he constructed his artificial language. However that may be, where a natural
language has sentences, and this holds of an interpreted language or an interpreta¬
tion of a formal language, Aristotle’s formal language does not have sentences per
se, but only formalized sentence patterns. A sentence possesses a truth-value; a
sentence pattern does not. Indeed, just as with a modern formal language, Aristo¬
tle’s artificial language is not strictly a language, since it contains no true or false
sentences. So, while his language is interpreted in respect of its logical constants,
it is not a fully interpreted object in respect of (1) its not being bound to a par¬
ticular universe of discourse and (2) its having schematic letters holding places for
terms that anticipate a given universe of discourse. In respect of the first point,
his language is formally applicable to every domain but is itself specific to none —
it is topic neutral. And in respect of the second point, ‘term’ is a metalinguistic
name for a formal part of a categorical sentence, that part which is filled by a
non-logical constant, a name or substantive. Aristotle’s definition of ‘term’ (opo<;)
in Prior Analytics AI is consonant with his practice.
I call that a term into which a premiss may be broken up, that is, both
that which is predicated and that of which it is predicated (whether or
not ‘is’ or ‘is not’ is added or divides them). (24bl6—18)
There are no terms in Aristotle’s formal language, only schematic letters holding
places for non-logical constants (terms); and, of course, a schematic letter is not
itself a term. The word ‘term’ (opoc;) exists in Aristotle’s metalanguage. In addi¬
tion, there are no logically true sentences in Aristotle’s syntax language. 40 Thus,
there are absolutely no truth-conditions for sentence patterns in Aristotle’s for¬
mal language. In principle, this is exactly the case with modern logics, save for a
logic involving identity and tautology. Aristotle had genuinely syntactic concerns,
although, again, without the sophistication and rigor of a modern mathematical
logician, but, nevertheless, with an intelligence sufficient for having accomplished
many of the same results as a modern logician. Aristotle’s formal language is
entirely a metalogical (metalinguistic) device used to objectify and exemplify, to
explicate, and to study his logic, and, moreover, it was conceived by him to be
applicable equally to all the axiomatic or axiomatizable sciences. 41
40 We might make one exception to this. While Aristotle’s syllogistic system seems to eschew
identity, he does cite some instances of categorical sentences with the same subject and predicate
terms. Surely Aristotle would recognize that in these cases it is impossible for such sentences to
be false, just as he recognized the impossibility of the compound sentence expressing the law of
the excluded middle, and any sentence expressing an instance of this law, being false. But such
sentences do not serve his scientific interests, and thus his logic is accordingly restricted. Cf.
n29.
41 Even a modern mathematical logician, when constructing a formal language, has some in-
144
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2.2 Truth conditions for object language sentences
In relation to establishing truth conditions for sentences, it is customary for a
modern logician to speak about ‘giving an interpretation’ of a formal language.
In this respect, then, a modern logician would (1) specify a non-empty domain
as a universe of discourse, (2) specify the meanings of all logical constants, (3)
establish definitions of ‘true’ and ‘false’, and (4) establish conditions under which
a given interpreted sentence is either true or false. However, in connection with
truth conditions, Aristotle did not employ a modern system of interpretations and
reinterpretations; he seemed not to work with model-theoretic notions. Thus, we
do not find him saying that “a given sentence is true under a given interpretation”.
Aristotle, however, does use a method of substituting non-logical constants for
schematic letters in categorical sentence patterns. And we do witness him establish
meanings for his logical constants and truth conditions for sentences, categorical
or otherwise. Above we treated their syntax and now we treat their semantics,
principally focussing on categorical sentences.
Since the four categorical sentence patterns are not themselves sentences pos¬
sessing a truth-value, we might wonder what conditions Aristotle required to pro¬
duce a categorical sentence according to his definition of ‘ logos' or ‘ apophansis ’,
that is, beyond his sentence formation rules. It is immediately apparent that he
would specify a universe of discourse — that is, he would introduce genuine non-
logical constants, or, what amounts to the same thing, he would apply his formal
language to a given domain. The following passage from Posterior Analytics A10
establishes that this is so.
Every demonstrative science [xacra diroSeiXTixf) emaxripr)] is concerned
with three things: [1] what it posits to exist [oaa xe itvat xtfiexai] (these
items constitute the kind [xo yevoc] of which it studies the attributes
[m0r)paxct>v] which hold of it in itself); [2] the so-called common axioms
[ra xotva Xeyopeva a^uopaxa], that is, the primitives from which its
demonstrations proceed; and [3] thirdly, the attributes [xa Kafir]] where
it assumes what each of them means [xl arjpatvet cxaaxov]. (76bll-16)
Aristotle also provided meanings for his logical constants. In addition, he defined
‘truth’ and ‘falsity’, which he understood to pertain to sentences, and he provided
the conditions under which a given sentence is true or false. We can take his dis¬
cussions, particularly in Posterior Analytics, on the genus of a science as evidence
tended interpretation in mind as pertains to variables, logical constants, and non-logical con¬
stants. When a logician identifies a notation designating variables, logical constants, non-logical
constants, he/she already has in mind sentence, derivation, and the meanings of the logical
constants. A logician never escapes language and indeed invents a formal language always an¬
ticipating its interpretations. Moreover, the distinction between logical syntax and semantics
is one that exists for the most part in thought only, and even in thought the distinction is not
complete.
Aristotle’s Underlying Logic
145
of the requirement that a non-empty domain be specified. In this section we exam¬
ine his definitions of the logical constants, his definitions of ‘true’ and ‘false’, and
the conditions under which a sentence is true or false. There also is a section on
Aristotle’s notion of existential import and a final note on his intensional notion
of truth. We begin with a brief statement on the importance he attributed to
meaning.
Aristotle on meaning in general
Aristotle gave special attention to the matter of meaning in various treatises.
In connection with semantics in general, he defined ‘having a meaning’ in rela¬
tion to intelligible discourse in Metaphysics 4-4, concerning the principle of non¬
contradiction, in the following, rather stipulative, way:
Suppose ‘man’ has the meaning ‘two-footed animal’. By ‘having a
meaning’ [to ev arjpaivsi] I mean this: if ‘man’ is ‘two-footed animal’,
then if anything is a man, its ‘being two-footed’ will be what its ‘being
a man’ is. (1006a31-34)
Aristotle recognized that the meanings assigned to words, and the words, or mean¬
ingful sounds themselves, are conventions. Accordingly, he expected those engaged
in intelligible discourse to agree that one word have one meaning, or if many mean¬
ings that this be made clear: “let us suppose ... that a word has a meaning and one
specific meaning [appdivov n to ovopa xai arjpaivov ev]” (Meta. 4-4'- 1006bl2-13;
cf. Meta. 11.5 : 1062all-23). Moreover, he made absolutely clear that we not
confuse a word with its denotation.
As to having a meaning, what we insist on is that the meaning is
not the object referred to [itself] (since then ‘musical’ and ‘white’ and
‘man’ could have a single meaning or referent, and all would be one,
and those terms would be synonymous). [What we mean is that] it will
not be possible to be and not to be the same thing [to ocjto], except
ambiguously; for example, if we call a ‘man’ what others were to call
a ‘non-man’. The question is not whether the same thing can at the
same time be and not be a man in name, but in fact [to 6’ ootopoupEvov
ou touto ecttiv, ei ev8ex eto(1 t o muto oipa iavai xai pf) feivai avhpcuTtov to
ovopa, txAAa to xpaypa]. (1006bl5-22)
Aristotle took up this matter throughout most of Metaphysics 4-4 , where he was
careful to state the necessity for clear definition and meaning in connection with
the law of non-contradiction. He wrote that “to signify its being means that its
being is not something else” (1007a26-27); and “there must, accordingly, be some
meaning in the sense of indicating a thing’s being” (1007bl6-17). Here again
Aristotle expected that words be used carefully and precisely in order better to
reflect in thought what exists independently of thought. Then, later in Metaphysics
4-1 he wrote that
146
George Boger
basic to all these argumentations [viz., eristic argumentations] are def¬
initions. And definition [optapoq] arises out of the necessity of stating
what we mean; for the sentence of which the word is a sign becomes a
definition [6 yap Xoyo<; ou xo dvopsa aqpiELOv 6ptap6<; eaxat]. (1012a21-
24)
Then, in connection with meaning in relation to truth and falsity, Aristotle wrote
in Metaphysics 4-8-
Against all such argumentations, however, it must be asked [as at Meta.
4-4 ■ 1006al8-22] ... not that something is or is not, but that something
has meaning [ou/i Elvod xt r) pit) eivai aXXa arjpidivEiv xi]; so that we must
converse on the basis of definition [e£ optapou] by grasping what falsity
and truth mean. (1012b5-8)
We shall treat truth and falsity more fully below, but note here that Aristotle is
quite clear about the importance of establishing meaning 42 and about the rela¬
tionship between meaning and existence. 43
Defining the logical constants
Aristotle provides some explicit definitions of his logical constants in Prior Ana¬
lytics and elsewhere. On ‘belonging to every’ and ‘belonging to no’ he writes in
Prior Analytics Al: “I call ‘belonging to every’ or ‘to none’ universal” (24al8).
He adds:
For one thing to be in another as a whole is the same as for one thing
to be predicated of every one of another. We use the expression ‘pred¬
icated of every’ when none of the subject can be taken of which the
other term cannot be said, and we use ‘predicated of none’ likewise.
(24b26-30)
In Posterior Analytics A4 he writes on universal predication in much the same
way:
42 On meaning, see Top. 1.5 & 7 .2-5 and Po. An. B13-14-
43 Cf. SR on confusing a word with an object it denotes. At the outset of SRI , where Aris¬
totle introduces the subject matter of fallacious argumentation, just after defining ‘refutation’
(eXeyx o ?)> he remarks that argumentations might appear to be cogent when they are not. He
then writes, setting the tone for what follows, that this might be due to “several causes, of
which the most fertile and widespread is the argumentation that depends on names. For, since
it is impossible to argue by introducing the actual things under discussion, but we use names as
symbols in the place of things, we think that what happens in the case of names happens also
in the case of the things, just as people who are counting think in the case of their counters”
(165a4-10).
Aristotle’s Underlying Logic
147
I say that something holds of every case if it does not hold of some cases
and not of others, nor at some times and not at others. For example,
if animal holds of every man, then if it is true to call this a man, it is
true to call him an animal too; and if he is now the former, he is the
latter too. (73a28-31; cf. A4 : 73b25-74a3 on ‘holding universally’ [‘to
xaGoXou])
In On Interpretation 1 he defines ‘predicating universally’ indirectly, when he
identifies some sentences as indeterminate. He writes:
It is necessary when asserting [dTtocpalvEaBca] as either belonging or
not belonging [&<; UTtdp)(£i it i] fjr)] sometimes to something universal
[xaBoXou] sometimes to an individual [xaB’ Exacrcov], Now, if someone
states universally of a universal that something belongs or does not
belong, there will be contrary sentences [evavnoa dirocpavaeu;]. I mean
by stating universally of the universal, for example, “Every man is
white” and “No man is white”. (17bI-6)
In Prior Analytics A1 when he treats ‘belonging to some’ and ‘not belonging to
some’ Aristotle writes in a rather eclipsed manner: “I call ‘belonging to some’,
‘belonging not to some’, or ‘belonging not to every’ partial [ev pcpsi]” (24al8-19).
He seems to have taken their meanings as evident to his audience. In addition,
Aristotle takes ‘belonging to some’ in two ways, implicitly in On Interpretation 1
but rather more explicitly in Prior Analytics. (1) In its determinate (Sicoplapevoc;)
meaning, ‘some’ means, as in ‘A belongs to some B ’, that ‘some Bs are A’ and
‘some Bs are not A’ but not that ‘possibly all Bs are A’. The determinateness of a
sentence pertains to its having only one meaning. A participant knows that, of a
given kind, some indeed have and some indeed do not have a given property. (2) In
its indeterminate (dSiopicrcoc) meaning, ‘some’ means ‘at least one, possibly all’.
Here a participant does not know, in the case of ‘A belongs to some B’, whether
some Bs are not A or every B is an A. He writes in Prior Analytics A1 that
I call belonging or not belonging without a universal or partial indeter¬
minate [dSiopiCTiov] as, for example, “The science of contraries is the
same” or “Pleasure is not a good”. (24al9-22)
Again, an indeterminate sentence is ambiguous. Aristotle does not usually cite
a partial sentence, that is, one specifically using a partial logical constant, to
identify indeterminateness. Rather, he usually cites a general sentence, such as
“Men are white” or “Pleasure is good”. In such cases he remarks that while ‘men’
is used universally, the sentence is indeterminate: it could mean “Some men are
white” and “Some men are not white” or “All men are white” (On hit. 7: 17a38-
17bl6). There he indicates the indeterminateness of “Man is white” and “Man is
not white”. He notes that
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George Boger
the universal ‘man’ is not used universally in the sentence. For the word
‘every’ does not signify the universal but that it is taken universally
[to yap roic; ou to xaGoXou arpalvei aXX’ oti xaGoXou]. 44 (17bl 1-12)
This matter is clearly stated in Prior Analytics A27 where he counterposes a
sentence being determinate to its being indeterminate. There Aristotle comments
on developing argumentational skills and instructs his students to select things that
follow the subject as a whole since “a syllogism is through universal premisses”.
He continues this thought and thereby clarifies the meaning of ‘indeterminate’:
Now, if it is indeterminate [dSioplcrcou], it is unclear [a&r)Xov] whether
the premiss is universal [xaGoXou], whereas if it is determinate this is
evident [Sicoptopevou 8e cpavepov]. (43bl4-15; cf. Pr. An. A4: 26b21-
25)
He also reveals his understanding when he establishes the inconcludence of a few
patterns of premiss pairs by “proving it from the indeterminate” . In Prior An¬
alytics A4, in connection with showing that a pair of premisses — the major a
universal attributive or privative, the minor a partial privative — does not result
in a syllogism, Aristotle notes that this must be established from the indetermi¬
nate. He writes:
Moreover, since “B does not belong to some C” is indeterminate, that
is, it is true if B belongs to none as well as if it does not belong to
every (because it does not belong to some), ... (26bl4-16; cf. Pr. An.
A5: 27bl6-23)
In this way, then, Aristotle defined his logical constants.
Finally, in this connection, Aristotle distinguishes variously between kinds of
declarative sentence (apophansis). There are, first, the affirmation ( kataphasis)
and the denial ( apophasis ). Using Aristotle’s terminology developed in Prior An¬
alytics, we can take a kataphasis to be a positive, or attributive (xaTryfopixot;),
sentence and an apophasis as a negative, or privative (aTeprjTLxot;), sentence that
uses a negative operator. Second, a sentence can be singular (xa9’ exacrrov), par¬
tial, or particular (ev pepei; xara pspoc;), or universal (xaGoXou). The first of these
determinations usually captures the quality of a sentence, the second its quantity.
Third, a sentence can be either determinate (Stcopiopevov or SiopioTo:;) or inde¬
terminate (dSiopioTOv), as we noted above. These distinctions are more sharply
defined in Prior Analytics than they are in On Interpretation, but the two works
are generally in accord on these sentential determinations. 45 We shall examine the
kinds of sentence more fully when we treat their truth-values in relation to the
matter of existential import.
44 In On Int. 17bl2-16 Aristotle rules out such sentences as “Every man is every animal”.
Aristotle takes a universal to be a secondary substance and not an individual. Later logicians
considered ‘taking a term universally’ to mean taking a term to be distributed.
45 Interestingly, Aristotle used ‘ huparchein 1 in both works, but formalizes its use in Pr. An. In
any case, we take his practice in both treatises as a reflex of his theory of substance.
Aristotle’s Underlying Logic
149
Defining truth and falsity
In On Interpretation 9 Aristotle treats the notions of truth and falsity especially
in relation to examining contrary and contradictory sentences. There he provides
definitions of ‘true’ [dXr)0fjc] and ‘false’ [(jteu&qc] that accord exactly with Alfred
Tarski’s treatment of the topic in “The concept of truth in formalized languages” . 46
Aristotle writes:
For, if it is true to assert [el yap dXqOsc e’uietv] that something is white
or not white, then it is necessarily [dvayxr) e)ivoct] white or not white.
And if it is [xoti el eaxt] white or not white, it was true to affirm or
deny it [dXr)0ec f)v cpavac f) ditocpvat]. And, if it is not [in fact] white [d
pf] uTtdpxei], then to say that it is will be false [cjieuSexaL]; if to say that
it is will be false [xal d ([rcuSerac], then it is not white [ouy GitdpxEi].
And so, it is necessary that the affirmation or the denial be true [coax’
avccyxr) xrjv xaxacpaaiv r] xf)v axocpaatv aXr)9rj £vocl]. (18a39-18b4)
The upshot of this discussion is to affirm that every declarative sentence is either
true or false and that “the truth of sentences consists in corresponding with states
of affairs [coaxe, exei opotwc ol Xoyoc aXqGecc coanep xa xpaypaxa]” (19a32-33).
In Categories 12 Aristotle writes in much the same vein, but he states the case
somewhat more strongly when he addresses various meanings of ‘prior’.
The existence of a man is reciprocal in relation to the true sentence
about him as it follows from there being [such] a man [xo yap dvcci.
dvGpcoixov avxcoxpecpec xaxa xf)v xoG dvat axoXouGqacv xpoc xovdXr)0f]
Ttepi auxoO Xoyov]. For if a man exists, then the sentence asserting
[6 Xoyoc cp Xeyopev] that a man exists will be true [dXrjGrjc]. And
conversely, if the sentence asserting [6 Xoyoc & Xeyopcv] that a man
exists is true [dXr)0f)c], then the man exists. The true sentence [6 psv
dXr)0f)c Xoyoc], however, is in no way the cause of the [given] state
of affairs [acxtoc xou dvat xo xpaypa]; and yet the state of affairs [xo
Ttpccypa] seems somehow to be the cause of the truth of the sentence
[tccoc dcxiov xoO icvoa aXr)0f) xov Xoyov]. For a sentence is called true
or false as the state of affairs exists or does not exist [xo yap dvac xo
Ttpccypa f) pf) aXr)0f|c 6 Xoyoc 0 c|ceu5^c Xeycxac]. (14bl4-22)
Aristotle is quite clear about distinguishing a sentence pattern from a sentence,
and a sentence from its denotation, or state of affairs, or even from its sense or
meaning. Again, we have ample evidence of this topic treated more fully through¬
out Sophistical Refutations and Rhetoric.
We can supplement what Aristotle writes on truth and falsity in these works
with what he writes in Metaphysics f.1-8 in connection with his discussion of the
laws of non-contradiction and excluded middle.
46 Tarski states his semantic definition in relation to natural language as follows: “a true
sentence is one which says that the state of affairs is so and so, and the state of affairs indeed is
so and so” (in Corcoran 1990: 155; cf. 154-165).
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George Boger
And the possibility of a middle between contradictories is excluded; for
it is necessary either to assert or to deny one thing of another [aXX’
dvdyxr) r] (pavoa r) cmocpavoa ev xa0’ evoc otiouv]. This is clear from the
definition of truth and falsity [SfjXov bk xptoxov pcv optaapsvoic xl to
aXr)0£<; xai c[ie0§oc]; for to deny what is or to affirm what is not is false,
whereas to affirm what is and to deny what is not are true; so that any
sentence that anything is or is not states either what is true or what
is false [to psv yap Xsysiv to ov pf) eivat. r] to pf) ov feivoa ([e05oc, to
Se to ov feivai xod to pf) ov uf) iivoa aXr)0£c, uote xai 6 Xsycrv ELvai r]
pf) aXr]0£uoEL rj (jiEUCTETai, aXX’ oote to ov XsyETai pf) Sivai. r) feivai oute
to pf) ov]. Hence, either what is is affirmed or denied, or else what
is not is affirmed or denied. There can be no middle ground. (4-7:
1011b23-29 & 11.6: 1063bl5-18; cf. On Int. 14 : 23b29-30)
We shall return to this matter in connection with Aristotle’s notion of existential
import.
Aristotle, then, is quite clear about ‘truth’ and ‘falsity’ applying to sentences
(logoi) and not to states of affairs ( pragmata ), which he characterized using ‘eivai’,
or ‘being [the case]’, and ‘pf) Eivai’, or ‘not being [the case]’. However, Aristotle
sometimes uses the words ‘ alethes ’ and ‘ pseudos ’ in relation to pragmata where we
would prefer using ‘is the case’ and ‘is not the case’ and thus would avoid making
a category mistake. Consider, for example, what he writes in Metaphysics 5.12
where he defines ‘possibility’ [§uvaxov] and ‘impossibility’ [aSuvaxov]. (Here we
transliterate, rather than translate, and mark in bold face, the Greek for ‘true’
and ‘false’ to objectify Aristotle’s meanings.)
[In the case where ‘impossibility’ means the opposite of ‘possibility’, the
impossible is] the contrary of what is necessarily alethes [to Evavxlov
ec; avayxrjc; aXr)0£<;]: that the diagonal of a square is commensurable
with its side is impossible, because that is something pseudos [oti
([eOSoc; to toioOtov], and its direct contrary, incommensurability, is
not only alethes [dXr)0ec;] but also necessary; that it is commensurable
is, therefore, not only pseudos [(]>e05o<;] but also necessarily pseu¬
dos [ec avayxrjc On the other hand, the contrary of this,
the “possible”, holds when it is not necessary for its contrary to be
pseudos [4»eu8oc]: it is possible for a man to be seated, for it is not of
necessity pseudos [[eOSoc] that he is not seated. The possible, then,
means: (1) what is not of necessity pseudos [to pf] e<; avayxrjc ([^Soc
arjpaivEi]; (2) what is alethes [to <xXt]0ec]; (3) what may be alethes
[to EvScyopcvov aXrjOcc Eivai]. (1019b23-33)
It is not uncommon for Aristotle, and he suggests that it is a common practice, to
use ‘ alethes’ and l pseudes ! to refer to both sentences (logoi) and states of affairs
(pragmata). In fact, he explicitly makes this point in Metaphysics 9.10 where he
remarks that being and nonbeing are commonly assessed according to the ‘true’
or the ‘false’, that is, by using the words ‘ alethes’ and ‘ pseudes He writes:
Aristotle’s Underlying Logic
151
This use depends on things being combined or dissociated [xouxo 8’
etu xcov xpaypaxcov ecru xw auyxeiaGcn i] 5tiQpfjcr9ot(.] ; so that he who
thinks that what is dissociated is dissociated, and what is combined is
combined, holds the truth, whereas he whose thought is contrary to the
state of affairs is in error [wctxe aXr)0£U£t psv 6 xo 8ir)pr)pEvov btopEvoc
§ir)pfjcr0ai xou xo auyxEtpEvov auyxEtoOaL, EtjjEuaxai 8e 6 Evavxiax; Eywv
r] xa xpaypaxa]. When, therefore, is there or is there not what is called
truth or falsity?
We must inquire into what we mean by this. For it is not because
we truly [dtXr)0w<;] hold you to be white that you are white; but it is
because you are white that we who assert this speak truly [aXr]9£uop£v].
(1051b2-9)
Here he uses l alethes’ and l pseudes’ to mean true and false in relation to sentences.
He also frequently uses, as he does here, the verbs ‘dXrjOEUEtv’ and ‘(|>eO§£a0ou’.
Continuing later in this same passage he writes:
As to “being” [xo slvat] in the sense of the true [ex; xo otXrjGep] and
“not being” [xo pf] elvou] in the sense of the false [ox; xo cpeOSop], there
are two cases: in one case there is truth [aXq0£<;] if the combination [ei
auyxELxai] [of subject and attribute] exists, and falsity [^eu8o<;] if there
is a dissociation [xo 8’ et pf) aoyxEixai]; in the other case, however,
whatever is, is as it is, or it is not at all. Here truth is the knowledge
of these things [xo Sc dXr)0e<; xo voeiv xauxa]. (1051b33-1052a2)
Aristotle also makes much the same point succinctly in Metaphysics 5.29.
The “false” [xo cpeOSoc;] refers (1) to a state of affairs as not the case
[ox; xpaypa ^eGSo;]: and this, on the one hand, because it is not put
together or cannot be put together. ... States of affairs [itpaypaxa] then
are said to be not the case [cJjeuSfj] whether because they themselves are
not or because the appearance derived from them is of something that
is not.
Next, (2) a false account, in so far as it is false, is the account of
things that are not. Hence, every account is false which is an account
of something other than that of which it is true [Xoyo<; 8e (Jjeu8?]c; 6 xwv
pf) ovxoiv, fj cjjsuSfjc, 8lo Tick; Xoyoq (ji£u8f)<; EXEpou r] ob Ecrxiv aXr]0f)c;];
for example, an account of a circle is false of a triangle. 47 (1024bl7-19,
24-28)
4 'Aristotle later writes that “just as we declare states of affairs to be ‘false’ which occasion a
‘false’ appearance” (1025a5-6). He refers in this passage to when the diagonal of the square is
said to be commensurate with the side as a ‘sense in which a state of affairs is not’ (1024b21).
We translate ‘adunaton’ in relation to ontic matters by ‘impossible’, in relation to sentences by
‘absurd’.
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We encounter an analogue of this ambiguity in English, where ‘true’ and ‘false’
have a range of meanings, including the genuine and the spurious. We might be
inclined to accuse Aristotle of making a category mistake, and perhaps there are
cases where he slips. However, it is obvious here that Aristotle does not commit a
category mistake, but that he uses the same word with two meanings corresponding
to the two contexts. He does not confuse a sentence (logos or apophansis ) with
a corresponding state of affairs (pragma), although we might wish that he had
always used different expressions to distinguish their being or not being. Thus, in
relation to states of affairs ( pragmata ), we could properly translate ‘alethes’ by ‘is
the case’ and ‘ pseudes' by ‘is not the case’.
Again, Aristotle is quite clear about pragmata either being or not being, ob¬
taining or not obtaining. This is something ontic. He affirms, in one sense, that
combination and division are mental acts, in particular, acts of predicating one
thing of another either attributively or privatively, which are participant relative.
In Metaphysics 6.4 he writes:
Now, being [to 8e <1k; a Xq0e<; ov] in the sense of being true and nonbeing
[xoa pf| ov <hc; ijjeuSoc;] in the sense of being false are concerned with
union [ouvOeotv] and division [Statpemv] and, taken together, with the
relation of contradictories [avxttpdaethc;]. For there is truth when an
affirmation corresponds to a combination in beings and when a denial
corresponds to a dissociation among beings; whereas there is error [or
nonbeing] when the opposite relations hold [to pev yap 6tXr)0e<; rqv
xaxazpcnv eiu tco auyxetpevw ex El T fi v S’ otrcoepaatv ext to SirjpiQpevo, to
§e iJjsuSoc; xouxou tou peptapou xf)v dvxlcpaatv]. (1027bl8-23)
A little later in this same passage he remarks that “the false and the true are not in
things [ou yap ecru to ^EoSoq xod to aXrjGet; ev xait; xpaypaatv], as if the good were
true and the bad were forthwith false; but they are in thought [aXX’ ev Siavola]”
(1027b25-27). We could not ask him to be more clear. He adds, nevertheless:
However, since unification and separation are in thought and not in
things, ‘being’ in this sense differs from ‘being’ in the chief sense. For
to predicate or deny what something is, or that it is of some sort, or
that it is so much, or the like requires thinking [f] Sidvota]. (1027b29-33)
And thoughts become ‘materialized’, or expressed, by means of sentences. Aristo¬
tle makes this point at Metaphysics ^.7 as follows:
Still, every concept and thought is expressed either as an affirmation
or a negation [exi xav to Siavorjxov xai vorjxov f) Siavola f] xaxckpamv
f] dxocpaatv]; this is clear from the definition [el; optapou] of truth and
falsity. When a sentence either asserts or denies, it expresses either
truly or falsely [oxotv dXqGeur) r) cJteOSrjxai] . 48 (1012a2-4).
48 This passage continues: “Whenever a sentence either asserts or denies [epaaa ij aTcotpaoa], it
expresses either truly [dXr)0Euei] or falsely [oxav 5 e £>5t, 4>eu5ETai.]” (1012a4-5).
Aristotle’s Underlying Logic
153
The distinctions that modern logicians believe themselves to have invented were
surely anticipated, if not invented, by Aristotle in ancient times. Categories helps
to make this even more evident.
Aristotle’s treatment of truth conditions in Categories
In Categories , as he sharpens the distinction between a sentence and its deno¬
tation, Aristotle defines truth and falsity by affirming that a sentence is not a
substance ( ousia ). In Metaphysics he makes it clear that a substance, or a subject
(hupokeimenon) , maintains the same identity and yet admits of contrary qualities.
In this connection he defines contraries, or contrariety, in Metaphysics 10.4 as
they pertain to substance. 49
Since things which differ may be more or less different, there is a great¬
est difference; this I call contrariety [evavxuomv]. We can show induc¬
tively that contrariety is the greatest difference. ... Hence the distance
between extremes is a maximum, and this constitutes the relation of
contrariety. ... From these considerations, then, it is clear that con¬
trariety is perfect difference. 50 (1055a3-6, 9-10, 16-17)
A given individual, then, at one time might be warm or good and at another
time cold or bad, but he/she cannot be both warm and cold or good and bad at
the same time in the same respect. For Aristotle a substance has a capacity for
contraries, but does not itself change identity. In Categories 5 he discusses the
mistake of thinking that a sentence (logos) or an opinion (56£oc [ doxa ]), which is
expressed by a sentence, admits of contrariety. He writes that
the same sentence [6 auxo<; Xoyoc] appears to be both true and false.
For example, if the sentence “Someone sits” is true, but if he rises,
the same sentence becomes false. And likewise with opinions [enl xfj<;
56£r)<;]. For if someone believes truly the sentence “Someone sits”, then
upon the person rising he will believe falsely if he still holds the same
opinion about him [xepi auxou Soqav]. (4a23-28)
49 In Meta. 5.10 Aristotle also writes that: “‘contrary’ means [1] attributes whose genera are
different and which cannot at the same time be present in the same thing; [2] things which differ
most in the same genus; [3] attributes which differ most in the same subject; [4] things which
diverge most from the same potentiality; and [5] things that differ widely either in themselves,
in genus or in form” (1018a25-31).
50 In Meta. 10.4 Aristotle continues by writing that: “all this being so, it is evident that one
thing can have only one direct contrary; a difference separates two things; therefore contrariety,
being complete difference, is a relation between two things. ... hence, a complete difference
between different things in the same genus is the greatest possible. We have shown also that
such complete difference is contrariety; for a complete difference is one that separates the species
of the same genus. ... for a single science covers a single genus and therefore deals with the
complete differences in that genus” (1055al9-20, 22-23, 26-29, 31-33). Cf. On Int. 14■ 23b22-
23. See also above §2.1 on Aristotle’s notion of universe of discourse.
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George Boger
Aristotle continues here to remark that whenever a substance admits of contrary
qualities it is due to a change within itself. However, in the case of a sentence
(logos) and an opinion ( doxa ) each
remains in itself unaltered in any and every respect ; but it is because of
a change in the fact [too 8e xpaypaToc; xivoupevou] that the contrary
applies to them. For the sentence “Someone sits” remains the same;
but according to changing conditions [xou Se xpaypaTop xivr)0£vxog] it
becomes at one time true and at another time false. As with sentences,
so too with opinions [mi rfjc; 8o£r)c;]. (4a34-4b2)
At 4b2-6 Aristotle asserts that it is the special property (I'Siov) of substance to
admit of changes (tiEiapoXf)) within itself, but that opinions and sentences do not
admit of such changes. He next forcefully states that anyone maintaining that a
sentence admits of contrary qualifications is speaking nonsense.
It is not because a sentence and an opinion [6 yap Xoyog xai f) 8o£a]
take on contrary qualities that they are said to take on contraries, but
because of what has happened to something else. For it is because
the fact is or is not that case [tm yap to xpaypa iivai rj [if) £ivai] that
a sentence [Xoyop] is called true or false, and not that it can itself
receive contrary qualities. For absolutely nothing [oacXwc; yap ooSev Ox’
ouSevop] can alter either a sentence or an opinion, and so, since they
cannot receive contraries nothing changes in them. (4b6-13)
Only substances can admit of such changes (4bl3-14). Here again Aristotle affirms
the difference between a sentence and the state of affairs denoted by the sentence
as he treats truth and falsity. In Categories 10 he develops this distinction with
even more precision.
Nor is what underlies [to uxb] an affirmation and a denial an affirmation
and a denial. An affirmation [xaxacpaau;] is an affirmative sentence
[Xoyoc; xaracparixoc;], a denial [axocpacnp] is a denying sentence [Xoyot;
axocparixop], But what underlies [0x6] either an affirmation or a denial
is not a sentence [ouSev ectti Xoyo<;]. Still, these things are said to
be opposed to each other as affirmation and denial; there is the same
manner of opposition. For just as an affirmation is opposed to a denial
— for example, “Someone sits” and “Someone does not sit” — so
are opposed the things that underlie each sentence [to Cep’ exaTEpov
xpaypa] — the sitting and the not sitting. (12b5-16)
As Aristotle had distinguished a word and its object, here he distinguishes a sen¬
tence from what it expresses. He clearly grasps the difference between a sentence
and its denotation. Later in this same discussion in Categories 10 he turns to a
position he treats in On Interpretation 7:
Aristotle’s Underlying Logic
155
It is evident that affirmations and denials are opposed in none of those
ways we have already treated. For only in relation to [contradictory]
sentences is it always necessary for one to be true the other to be
false. With contraries it is not always necessary for one to be true the
other false, nor with relatives, nor with possession, nor with privation.
For example, health and sickness are contraries, but neither the one
nor the other is either true or false; likewise with the relatives the
double and the half. Nor are privation and possession such as sight
and blindness. Generally, nothing that is said without combination is
either true or false [oX«<; §£ xtov xaxa prjSepiav aupiTtXoxfjv Xcyopevwv
ouSev oute aXr]0£c; oute c[;e086c; eoxlv], All the opposites just treated
are said without combination [avcu oupxXoxfic;]• (13a37-13bl2)
Thus, Aristotle distinguishes a sentence from its denotation, and he establishes
that the truth or falsity of a given sentence, which are ontic determinations, de¬
pends upon correspondence with the states of affairs or facts, which also are ontic
matters, denoted by the sentence as being the case (to Slvou) or not being the case
(xo [if] £tvai). 5i
Truth-values of contradictory and contrary sentences
In Metaphysics 5.10 Aristotle defines ‘opposite’ ( antikeimenon ) as having a va¬
riety of meanings: contradiction, contrariety, correlation, privation, possession.
What he writes there about being corresponds exactly with what he writes in
On Interpretation 7 having to do with contradictory and contrary sentences. In
Metaphysics 10.4 he states of substance that “the primary form of contrariety
[jtpdjxr] 5 e Evavxiwau;] is that between a positive state and a privation [ecu; xcd
axEpr]aic Eaxtv]” (1055a33-34). 52 He continues there to distinguish contradiction
from contrariety as follows:
Opposition [avxfxexat] may take the form of contradiction
[dvxtcpaatc;] or of privation [axEprjau;] or of contrariety [evavxioxrjc] or
of relation [jtpoc xl] . The first of these is contradiction, and contradic¬
tion admits of no intermediate, whereas contraries do; it is clear that
contradictories and contraries are not the same. (1055a38-1055b3)
Aristotle defines ‘contrary’ and ‘contradictory’ in Prior Analytics B15:
63b23-30 in a way that exactly comports with what he writes in On Interpre¬
tation , but in Prior Analytics in relation to the logical constants.
51 Below this section we take up whether meaning is an extensional or an intensional determi¬
nation according to Aristotle.
52 In Meta. 10.5 Aristotle writes on contraries that “it is rather an extreme which has something
between it and its opposite. In that case it must be an opposite either as a denial or as a privation.
It cannot be the denial or privation, for why would it be of the greater rather than of the less?
Hence, it must be the privative denial of both” (1056al4-18).
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I say that verbally there are four <pairs of> opposite sentences [npotaoei?],
to wit: [1] ‘to every’ and ‘to no’, [2] ‘to every’ and ‘not to every’, [3] ‘to
some’ and ‘to no’, and [4] ‘to some’ and ‘not to some’. In truth, how¬
ever, there are three, for ‘to some’ and ‘not to some’ are only opposites
verbally. Of these, I call the universal sentences contraries [evavTioi?
pev to ic, xaOoXou] (‘to every’ is contrary to ‘to none’, as, for example,
“Every science is good” is contrary to “No science is good”) and the
other pairs of sentences opposites [xa<; S’ aXXat; avxiXEipcvac]. (63b23-
30)
Only opposites cannot belong to the same thing at the same time. According to
the principle of opposition, both opposites obtaining at the same time is impossible
(Meta. 10.5 : 1055b37-1056a3). When he turns in On Interpretation 7 to treat
sentences, he states that “the denial ... must deny the same thing the affirmation
affirms of the same thing” (17b39-40). 53 And there also he makes this point about
contraries: “but what constitutes sentences as contrary is having two contrary
meanings, not having two contrary subjects” (23b6-7). And, “it is impossible for
opposite sentences [xa<; dvrixeipevac tpaaetc] to be true about the same thing”
(21bl7-18). In fact, when one of a pair of contradictories is true the other is
necessarily false. This is not the case with contraries where both might be false,
but not both true. In On Interpretation 1 he writes:
But I call the universal affirmation [xrjv too xaGoXou xoa&cpacnv] and
the universal denial [xai xf)v xoO xaGoXou aTtocpacuv] contrarily opposite
... Hence, these cannot be true together, but it is possible that their
opposites [i.e., sub-contraries] can be true of the same thing. (17b20-
26)
Among the relationships Aristotle understands to exist between categorical sen¬
tences are two pairs of contradictories and one pair of contraries, all of which
he employs in his deduction system. Now, rather than grasping some syntactic
relationships among categorical sentences as taking a given sentence and then as
a result being able or not able to take another sentence, Aristotle treats their
semantic relationships as follows.
For contradictories:
1. If a sentence fitting the pattern AaB is true, then a sentence fitting the pat¬
tern AoB is necessarily false; if a sentence fitting the pattern AaB is false,
then a sentence fitting the pattern AoB is necessarily true.
If a sentence fitting the pattern AoB is true, then a sentence fitting the pat¬
tern AaB is necessarily false; if a sentence fitting the pattern AoB is false,
then a sentence fitting the pattern AaB is necessarily true.
S3 This passage continues: “whether of something partial or universal, taken as universal or as
not universal” (17b40-18al).
Aristotle’s Underlying Logic
157
2. If a sentence fitting the pattern AeB is true, then a sentence fitting the
pattern AiB is necessarily false; if a sentence fitting the pattern AeB is
false, then a sentence fitting the pattern AiB is necessarily true.
If a sentence fitting the pattern AiB is true, then a sentence fitting the
pattern AeB is necessarily false; if a sentence fitting the pattern AiB is
false, then a sentence fitting the pattern AeB is necessarily true.
For contraries:
1. If a sentence fitting the pattern AaB is true, then a sentence fitting the
pattern AeB is necessarily false; but if a sentence fitting the pattern AaB
is false, then a sentence fitting the pattern AeB is not necessarily true but
might be false.
2. If a sentence fitting the pattern AeB is true, then a sentence fitting the
pattern AaB is necessarily false; but if a sentence fitting the pattern AeB
is false, then a sentence fitting the pattern AaB is not necessarily true but
might be false.
Aristotle states all this in so many words, although not as rigorously, and it is
obviously an important part of his treatment of sentences in Prior Analytics. This
position is explicitly treated in On Interpretation and it is used throughout his
analyses in Prior Analytics, notably there in relation to conversion ( A2 : 25al7-19)
and to his treating reductio proofs (e.g., those of Baroco in A5: 27a30-27b3 and
of Bocardo in A6: 28bl7-20).
Aristotle on existential import
Aristotle noticed some difficulty concerning the semantics of some sentences, par¬
ticularly indeterminate sentences and those having to do with future events (On.
Int. 9). 54 Still, his considerations of these matters in On Interpretation (6, 7-8,
9-11, 12, 13) seem well resolved. He reaffirms that every sentence is either true or
false, although determining this in one or another case may be difficult or some¬
times impossible. Modern logicians, however, seem more puzzled by considerations
of existence, and they fear that Aristotle’s logic leads to peculiar violations of the
54 There has been much discussion about ‘the sea-battle episode’ in On Int. 9, where Aristotle
seems to question the truth that every sentence is either true or false. In truth Aristotle refutes
a sophistic argumentation - one that aims to establish an ontology of predestination as following
from the truth of the law of excluded middle as applied to both sentences and states of affairs -
by reaffirming that “it does not make any difference whether any people made the contradictory
statements or not. For clearly this is how the actual things are even if someone did not affirm it
and another deny it. For it is not because of the affirming or denying that it will be or will not
be the case, nor is it a question of ten thousand years beforehand rather than any other time”
(18b36-19al). Here again Aristotle distinguishes between what is the case and knowing what is
the case. Thus, he always preserves the principle that a given sentence is either true or false.
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square of opposition, particularly in relation to contradictories. 55 However, this
matter was not especially troubling to Aristotle. In Categories 10 he addresses
this matter in the following way.
It is evident that affirmations and denials are opposed in none of those
ways we have already treated. For only in relation to [contradictory]
sentences is it always necessary for one to be true the other to be
false. With contraries it is not always necessary for one to be true the
other false, nor with relatives, nor with possession, nor with privation.
For example, health and sickness are contraries, but neither the one
nor the other is either true or false; likewise with the relatives the
double and the half. Nor are privation and possession such as sight
and blindness. Generally, nothing that is said without combination is
either true or false [oXox; 8e rnrv xaxa prjSepiav auprtXoxqv XeyoupEvcnv
ouSev ouxe dXrjOec; outs c|>£u86<; ecsxlv]. All the opposites just treated
are said without combination [otveu 0Ufi7iXoxf)<;].
However, it might seem that some such thing follows in the case of con¬
traries said with combination [xorra aupxXoxqv] — [as in] the sentence
“Socrates is ill” is contrary to “Socrates is well”. Yet, even in these
cases it is not always necessary that one sentence be true and the other
be false. For, if Socrates exists , one is true and the other is false; but if
Socrates does not exist , both [sentences] are false. For neither will the
sentence “Socrates is ill” nor the sentence “Socrates is well” be true if
Socrates himself does not exist.
As for affirmations and denials [era §e ye Tifjc xctxc^daeax; xai xrjc;
dmocpaaEOx;], if [the subject] does not exist, then neither sentence is
true. But if [the subject] exists, even then one or the other will not al¬
ways be true. The sentence “Socrates has sight” is the opposite of the
sentence “Socrates is blind” [in the sense in which ‘opposite’ is applied]
to privation and possession. For, if he [viz., Socrates] exists, it is not
necessary that the one sentence be true and the other false (since until
the time when it is natural for him to have sight both sentences are
false). While if Socrates does not exist then both sentences are false:
both “He has sight” and “He is blind”.
However, concerning affirmation and negation [i.e., contradictories] the
one will always be false the other true whether or not [the subject] ex¬
ists. For take the sentence “Socrates is ill” and the sentence “Socrates
is not ill”, if he exists it is evident that the one or the other must be
true or false. It is the same if he does not exist. If Socrates exists,
the sentence [expressing] that he is sick is false, but the sentence [ex¬
pressing] that he is not sick is true. Thus, it is characteristic [iSiov] of
55 On the topic of existential import and related matters see: A. Church 1964; R. M. Eaton
1959: 157-234; Kneale and Kneale 1962: 45-67; Cohen and Nagel (Corcoran 1993): 41-68; W. T.
Parry k E. Hacker 1991: 179-185; W. T. Parry 1966; I. Copi 1986: 177-193; and J. Lukasiewicz
1958: 59-67.
Aristotle’s Underlying Logic
159
these only — sentences opposed as affirmation and denial [viz., contra¬
dictories] — that the one is always true and the other always false in
all cases will hold of those opposites only which are in the same sense
opposed as affirmative and negative sentences. (13a37-13b35)
We cite here again a passage from Categories 12 where he writes about truth and
falsity.
The existence of a man is reciprocal in relation to the true sentence
about him as it follows from there being [such] a man [to yap iiivoa
avGpwxov avxiaxpEipEi xaxa xfjv xou aval axoXouGrjaiv xpog xov dXr)Gf)
tie pi auxoO Xoyov]. For if a man exists, then the sentence [6 Xoyoc]
asserting that a man exists will be true. And conversely, if the sentence
asserting that a man exists is true, then the man exists. The true
sentence, however, is in no way the cause of the [given] state of affairs
[to xpotypa]; and yet the state of affairs [to xpaypa] seems somehow to
be the cause of the truth of the sentence. For a sentence is called true
or false as the state of affairs exists or does not exist. (14bl4-22)
Aristotle affirms that the truth or falsity of a given sentence depends, first and
foremost, upon whether what it expresses corresponds to a given state of affairs.
In this connection, then, it depends upon whether the objects denoted by the
subject exist or do not exist, whether what is asserted is the case or is not the
case. The cause, or ground, of the truth of a sentence is the state of affairs it
denotes. This notion is underwritten by his notion of substance. “Were there no
individuals existing of whom it could thus be affirmed, it could not be affirmed of
the species; and were there no primary substance, nothing else could so much as
exist” (Cat. 5: 2a38-2bl). Aristotle has remarked in what way existence is the
cause of a sentence being true (Meta. \.1\ 101 lb23-29).
With what he writes in Categories 10 and 12 and elsewhere on truth and falsity,
we can make sense of the semantics of the various sentences that Aristotle treats
concerning their existential import. In general, in the case of existence, a sentence
is true or false as, respectively, the state of affairs denoted by the sentence is the
case or is not the case; in these cases there are no empty classes. In the case of
non-existence, no affirmation is true because it affirms something to be the case
that is not the case, and every privative, that is, every sentence with a negative
operator, is true because it truly expresses what is not the case.
Below we set out Aristotle’s semantics according as he considers sentences to
be: (1) singular (xa0’ Exaoxov), universal (xaGoXou), or partial (ev jaepei; xaxa
pspoc;); (2) attributive (xaxrjyopixo^; e^u;) or privative (axEprjxixoc;) — affirma¬
tive [positive] or negative: an affirmation (xaxdcpacnc;) or a denial (ouiocpocaK;);
(3) determinate (Siopurrop) or indeterminate (dSiopiaxop); (4) having a subject
that exists or having a subject that does not exist. Aristotle understands an op¬
posite (dvTixstpevov) of a given affirmative sentence to be either (1) a contrary
(evavxlov), which may or may not involve a negative operator, or (2) a contradic¬
tory (avxlcpaaic;), which always involves a negative operator. In our treatment of
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George Boger
sentences below we always take a sentence to be a simple sentence according to
Aristotle’s stipulation in his formal language.
The singular sentence
A singular sentence predicates, attributively or privatively, one thing of a particular
this , a primary substance (icpwrr) ouatct), and this particular is not predicable
of anything else {On Int. 7: 17a38-17bl; Cat. 5: 2all-14; Pr. An. A27:
43a25-29, 39-40). Every singular sentence is determinate. The opposite of a given
attributive singular sentence is either its contrary or its contradictory. A contrary
of an attributive singular sentence does not have a negative operator and is always
another attributive singular sentence (sc. an affirmation). Since, for Aristotle,
every denial involves a negative operator, there are no privative contraries in the
case of singular sentences. Using a negative operator (‘[if)’ or ‘ou’) in the case of an
attributive singular sentence, either adverbially as attached to a verb, or logical
constant, or as prefixing an entire sentence, always results in the contradictory
of the given singular sentence. Prior Analytics treats singular sentences only
incidentally. Moreover, Prior Analytics does not prefix a given sentence with ‘ ou ’
to produce its negation. Table 7 represents Aristotle’s thinking on the semantics
of singular sentences. 56
The universal sentence
A universal sentence predicates, attributively or privatively, one thing of every
or of no member of a given kind, a secondary substance (SeOtEpa ouata). Every
universal sentence is determinate. The opposite of a given attributive universal
sentence is either its contrary or its contradictory. A contrary of an attributive
universal sentence might or might not involve a negative operator, as in both On
Interpretation and Prior Analytics. The negative operator in these cases appears
as a pronominal adjective modifying the subject (or as part of the logical constant,
which nevertheless modifies the subject). The contradictory of a given universal
attributive sentence involves a negative operator in both On Interpretation and
Prior Analytics. The negative operator in these cases might appear as prefixing an
entire sentence, as in On Interpretation but not in Prior Analytics , or adverbially
as attached to a verb as part of the logical constant. Table 8 represents Aristotle’s
thinking on the semantics of universal sentences.
The partial sentence
A partial sentence predicates, attributively or privatively, one thing of some or of
not every member of a given kind. Here there are instances of both determinate
and indeterminate sentences. General sentences, for example, “Man is white”
or “Pleasure is good”, lack a universal quantifier and thus can be interpreted as
denoting both some and all; their meaning is not determinate but ambiguous.
56 In Tables 7-9 the ‘I’ pertains to On Interpretation and the ‘2’ to Prior Analytics', ‘T’ =
true and ‘F’ = false.
Aristotle’s Underlying Logic
161
Table 7.
Semantics of Singluar Sentences
Given affirmation
Opposites
Contrary
Contradictory
1. Socrates ails.
2. Ailing belongs to
Socrates.
1. Socrates fares.
2. Faring belongs to
Socrates
1. Socrates does not ail.
Not - Socrates ails.
2. Ailing does not be¬
long to Socrates.
Corresponding
truth values
Existent subject
When an affirmation is
T
its contrary is F.
its contradictory is F.
When an affirmation is
F ...
its contrary may be T
or F.
its contradictory is
T.
Non-existent sub¬
ject
No affirmation is T.
Every affirmation is F.
No affirmation is T.
Every affirmation is
But every denial is T.
When an affirmation is
F...
its contrary is F.
its contradictory is
T.
The opposite of a given determinate partial attributive sentence is either its sub¬
contrary (as modern logicians name it) or its contradictory. A sub-contrary of a
given partial attributive sentence might, as in both On Interpretation and Prior
Analytics , or it might not involve a negative operator. The negative operator in
the case of a sub-contrary appears adverbially as attached to a verb, or as part of a
logical constant. The contradictory of a given partial attributive sentence involves
a negative operator in both On Interpretation and Prior Analytics. The negative
operator in the case of a contradictory may appear as a pronominal adjective as
part of the logical constant, or as merely modifying the subject, or as prefixing an
entire sentence as in On Interpretation. Table 9 represents Aristotle’s thinking on
the semantics of partial sentences. 57
Denying a given privative sentence does not seem to have been treated by Aris¬
totle in either On Interpretation or Prior Analytics, although there are suggestions
in On Interpretation. Because he lacks a notion of double negation, or, at least,
it seems, a strong notion of double negation, Aristotle does not treat denying a
privative sentence save for reverting to its already given affirmation. This is par¬
ticularly true in Prior Analytics , where he does not negate an entire sentence with
‘on’ as in On Interpretation. Rather, there he begins with an affirmation, then
57 While Aristotle does not have an expression for ‘sub-contrary’, he does recognise their exis-
tence: On Int. 10 at 20al6-23. In Table 9 we interpolate to complete Aristotle’s thinking.
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Table 8.
Semantics of Universal Sentences
Given affirmation
Opposites
Contrary
Contradictory
1. Every man is good.
2. Good belongs to every
man.
1. Every man is bad.
No man is good.
2. Bad belongs to every
man.
Good belongs to no
man.
1. Not - every man is
good.
2. Good does not be¬
long to some man.
Good does not be¬
long to every man.
Corresponding
truth values
Existent subject
When an affirmation is
T
its contrary is F.
its contradictory is F.
When an affirmation is
F ...
its contrary may be T
or F.
its contradictory is
T.
Non-existent sub¬
ject
No affirmation is T.
Every affirmation is F.
No affirmation is T.
Every affirmation is
F.
But every denial is T.
But every denial is T.
When an affirmation is
F...
its contrary may be T
or F.
its contradictory is
T.
provides its contradictory, and then looks back at the original affirmation to ob¬
tain the contradictory of the negation: he does not, then, negate the negation to
obtain its own contradictory. As he remarked at different places, each affirmation
has its own, one negation, each negation its own, one affirmation — and he takes
them as pairs.
Finally, in respect of existential import, according to his definitions of true
and false, Aristotle holds that “to deny what is or to affirm what is not is false,
whereas to affirm what is and to deny what is not are true; so that any sentence
that anything is or is not states either truly or falsely” {Meta. ^.7: 1011b26-28).
This notion is again poignantly expressed in Metaphysics 5.7:
‘Being’ [to feivai] means [arjpouvEx] the ‘true’ when something is the
case and ‘not being’ [to pf) feivou] <means> the ‘not true’ but the
‘false’ when something is not the case; likewise for affirmation and
denial [exi xaTacpaaeux; xai axcKpacrecoc;]; for example, that “Socrates is
musical” means that this is the case <but we know that this sentence
is false>, or that “Socrates is non-white” means that this is the case
Aristotle’s Underlying Logic
163
Table 9.
Semantics of Partial sentences
Given affirmation
Opposites
[Sub-Contraries]
Contradictory
1. Some man is good.
2. Good belongs to some
man.
1. Some man is bad.
Some man is not
good.
2. Good does not be¬
long to some man.
Bad belongs to some
man.
1. Not - some man is
good.
No man is good
2. Good belongs to no
man.
Corresponding truth
values
Existent subject
When an affirmation is T
When an affirmation is F
its sub-contrary may
be T or F.
its sub-contrary is T.
its contradictory is F.
its contradictory is
T.
Non-existent subject
No affirmation is T.
Every affirmation is F.
No affirmation is T.
Every affirmation is
F.
Every denial is T.
But every denial is T.
When an affirmation is
F...
Every affirmation is
F.
Every denial is T.
its contradictory is
T.
<but, again, we know that this state of affairs is not the case>. But
that “the diagonal <of the square> is not commensurate <with the
side>” means that this is not the case <and this, of course, is not the
case, and so the sentence is true>. (1017a31-35)
Thus, for Aristotle, every sentence affirming something of a non-existent subject is
false because it affirms, incorrectly, something to be the case that is not the case. In
addition, every sentence denying — that is, using a negative operator — something
of a non-existent subject is true because it affirms, correctly, something not to be
the case that indeed is not the case. The correspondence, or non-correspondence
as the case may be, of thought to being is foundational in Aristotle’s thinking
about truth and falsity. Now, when we turn back to On Interpretation 6, just
after where he writes that an affirmation states something of something, a denial
denies something of something, we can better understand Aristotle’s meaning.
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Now it is possible to state of what does belong that it does not belong,
of what does not belong that it does belong, of what does belong that
it does belong, and of what does not belong that it does not belong
[eitEL 8e ectti xai to unapyov dmocpoaveaGoa wp jif) uxapyov xai to pf)
unapyov cix; uxapyov xai to uxapyov <2><; ujtapyov xai to pf| uitapyov dx;
pf] uxapyov]. (17a26-29)
When someone asserts of what does not belong, or is not the case, that it does
not belong, or is not the case, he/she speaks truly.
Truth-value: an extensional or intensional determination?
Considering the meanings Aristotle assigns to his logical constants might lead one
to believe that the truth-value of a categorical sentence is determined extensionally.
Recall, for example, that he defines ‘belonging to every’ by writing that “for one
thing to be in another as a whole ... ”. This suggests his taking the application of
the universal attributive logical constant extensionally with respect to non-empty
domains. However, Aristotle continues this statement by writing that .. is the
same as for one thing to be predicated of every one of another”, or “to hold in
every case”. His writing here seems shy of an extensional determination.
Now, while interpreting the relationship of terms in a categorical sentence as
that between classes, or even of sets, has been fruitful, this interpretation, never¬
theless, does not reproduce Aristotle’s own understanding. It has become a com¬
mon practice to define the truth of a categorical sentence in this way as follows
(using traditional expressions; ‘A’ and ‘B’ are placeholders).
“Every A is a B” is true iff the extension of A is entirely included in
the extension of B.
“No A is B” is true iff the extension of A is disjoint with the extension
of B.
“Some A is B” is true iff the extension of A intersects with the exten¬
sion of B.
“Some A is not B” is true iff the extension of A is not entirely included
in the extension of B.
Perhaps in some cases Aristotle did envisage terms to relate extensionally as
classes. Nevertheless, this interpretation does not take into account Aristotle’s
notions of attribution and privation, which pertain to things ( pragmata ), and his
notions of predicating of and not predicating of, which pertain to sentences (apo-
phanseis). Underlying his theory of predication in Categories, On Interpretation ,
and in Prior Analytics is his theory of substance in Categories and Metaphysics.
Briefly, for Aristotle attribution is ontic and independent of a participant, while
predication is intentional or linguistic and participant dependent. Predicating is
an activity of a human being reflecting in thought what exists or does not exist
outside of thought. For Aristotle, insofar as he considers matters of logic, thought
Aristotle’s Underlying Logic
165
follows being. And his notion of substance ( ousia) is precisely that in which prop¬
erties ( pathemata ) inhere or do not inhere. A substance, whether primary or
secondary, has or does not have one or another property. This explains Aristotle’s
use of ‘ huparchein ’ in the logical constants.
In order better to grasp term relationships and to see Aristotle as not deter¬
mining truth extensionally, we might consider his objection to platonic forms. For
Plato, an individual person’s being two-footed, for example, is just his participa¬
tion in the transcendent form ‘two-footedness’. Aristotle, rejecting the reality of
such transcendent forms and their putative explanatory value, rather thought of
an individual person’s being two-footed as his having this property or attribute
— as this characteristic inhering in, or as immanent in, a subject. A sentence ex¬
pressing this relationship would be attributive. To express the notion of a horse,
on the other hand, as not having rationality would be privative. Aristotle treats
attribution, in so far as a human expresses attribution in thought by means of
sentences, rather fully in Categories as well as at places in Metaphysics.
Since his theory of substance underlies his theory of predication, we ought rather
to say that truth for Aristotle is determined intensionally (or, perhaps, ‘posses-
sionally’), remembering his correspondence notion of knowledge. Thus, (using his
sentential expressions) we have the following (see nl7):
“A belongs to every B” is true iff every individual B has property A.
“A belongs to no B” is true iff no individual B has property A.
“A belongs to some B” is true iff some individual B has property A.
“A does not belong to some B” is true iff some individual B does not
have property A.
Aristotle uses ‘ hupokeimenon' to refer to the subject of a categorical sentence,
and he uses l pathe’ or ‘ idion ’ to refer to a property attributable or not attributable
to a subject. ‘ Pathe ’ contains a notion of ‘affect’, and surely not a notion of ‘class’,
but of something ‘happening to’ an individual this. Indeed, again, this is just his
meaning of the categories (Cat. 4)- If we add, as Aristotle sometimes did, that
property A is essential and not accidental to subject B, then we understand him
to mean that having property A, or being A, is just what it means to be B. We
again cite a passage from Metaphysics tersely to illustrate his thinking.
Suppose ‘man’ has the meaning ‘two-footed animal’. By ‘having a
meaning’ I mean this: if ‘man’ is ‘two-footed animal’, then if any¬
thing is a man, its ‘being two-footed’ will be what its ‘being a man’ is.
(1006a31-34)
Thus, we can take the following expressions of the universal attributive logical
constant (and the corresponding sentence patterns) to be generally equivalent in
meaning for Aristotle.
“A belongs to every B.” “A holds of every B.” “Having property A
belongs to every B.” “Being A is a property of every B.”
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2.3 The deduction system of Aristotle’s underlying logic
In Prior Analytics Aristotle turned his attention away from object language dis¬
courses and toward objectifying the formal deduction apparatus used to establish
scientific theorems. He was especially concerned to determine “how every syllogism
is generated” (25b26-31). He refers here to the elements of syllogistic reasoning,
which consist in elementary two-premiss valid arguments — the syllogisms — that,
when chained together, make up longer syllogistic (deductive) discourses. Aristo¬
tle’s project was to identify all the panvalid patterns of such elementary arguments.
In this way he explicitly treated deduction rules and their logical relationships in
Prior Analytics. These rules include both the one-premiss conversion rules and
the two-premiss syllogism rules. He accomplished this by exhaustively treating
each and every possible argument pattern relating to both the one-premiss and
two-premiss rules: (1) those patterns of arguments having a premiss-set of just
one categorical sentence with only two different terms — the conversions; and
(2) those patterns of arguments having a premiss-set of just two categorical sen¬
tences with only three different terms — the syllogisms. Aristotle limited his study
of multi-premiss arguments to their two-premiss patterns because two categorical
sentences, taken together, have ‘the fewest number of terms and premisses through
which something different than what was initially taken follows necessarily’. 58 The
syllogisms, then, are the building blocks of longer deductive discourses.
We have thus far examined the grammar of categorical sentences as Aristotle
treated this matter with the artifice of his formal language. We have also treated
their semantics. We turn now to the syntactic matter of generating, or transform¬
ing, sentences from given categorical sentences according to stipulated rules. This
defines the deduction system of Aristotle’s underlying logic. Briefly, the deduction
system as presented in Prior Analytics A1-2, 1,-6 consists in the following (Table
10 :
Table 10.
Aristotle’s deduction system
1. Four kinds of categorical sentence.
2. Two pairs of contradictories.
3. One pair of contraries.
4. Three one-premiss deduction rules: the conversion rules.
5. 14 two-premiss deduction rules; the syllogism rules.
6. Two kinds of deduction: direct, or probative deduction, and
indirect, or reductio, deduction.
Below we extract the rules Aristotle used for forming one-premiss arguments and
the three corresponding conversion rules and his rules for forming two-premiss
arguments and the corresponding syllogism rules.
58 The text here is a gloss. See Pr. An. B2: 53bl8-20 and Po. An. A3\ 73a7-ll Bll: B24-27.
Also see the definition of ‘ sullogismos’ in Pr. An. Al: 24bl8-20.
Aristotle’s Underlying Logic
167
The one-premiss conversion rules
In Prior Analytics A2 Aristotle treats converting (to oiVTiaxpecpeiv; conversion:
avTiCTiporpr)) the predicate and subject terms of categorical sentences to extract
certain deduction rules. To do this he treats each of the four kinds of categorical
sentence metalogically; that is, he treats at one time, say, all universal privative
sentences, by treating the one sentence pattern that they all fit. Aristotle models
an object language conversion as an elementary one-premiss argument pattern —
where a given sentence is in the role of premiss and its conversion is in the role
of conclusion. In this way, without considering object language arguments but
only their patterns, he determines which sentences logically convert with respect
to terms and which do not by establishing that the argument pattern that they
fit is panvalid. Aristotle determines that of the four kinds of categorical sentence
three logically convert and thus their panvalid argument patterns can serve as
deduction rules. Interestingly, he first states each of the three conversion rules in
a sentence (25a5-13), as we would expect a rule to be expressed. Moreover, he
treats the conversions in Prior Analytics A2 exactly as he does the syllogism rules
in Prior Analytics A4-6 (§3.2 and n68). Aristotle assumes his reader’s familiarity
with converting, since he does not define conversion per se: he takes it as obvious
that conversion involves changing the places of the subject and predicate terms
in a given categorical sentence while leaving its logical constant unchanged (save
for per accidens conversion). Aristotle considers these formal transformations
to be deduction rules because something different than what is initially taken is
established to follow necessarily in each instance. Their being universal in this
respect underlies their rule nature.
Aristotle does not explicitly state any general syntax rules for forming premiss-
sets or for forming premiss-conclusion arguments. Still, in this connection he tends
to be more explicit in the case of the syllogisms than in the case of the conversions.
In any case, we can easily extract from his practice of treating conversion in Prior
Analytics the following conversion premiss formation rules.
CPFR1 A conversion premiss-set consists in one and only one of any of the four
categorical sentences.
This rule could be generalized for any one-premiss argument.
CPFR2 The two non-logical constants in the categorical sentence in a conversion
premiss-set are not identical.
CPFR3 Nothing is a conversion premiss-set except in virtue of these rules.
The following are one-premiss argument formation rules implicit in Aristotle’s
treatment of conversion.
CAFR1 A one-premiss conversion argument consists in one and only one cate¬
gorical sentence in the role of premiss and one and only one categorical
sentence in the role of conclusion.
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George Boger
CAFR2 The logical constant in the conclusion sentence is the same as the logical
constant in the premiss sentence. 09
CAFR3 The non-logical constant in the predicate position in the conclusion sen¬
tence is the same as the non-logical constant in the subject position in
the premiss sentence, and the non-logical constant in the subject position
in the conclusion sentence is the same as the non-logical constant in the
predicate position in the premiss sentence.
CAFR4 Nothing is a one-premiss conversion argument except in virtue of these
rules.
We can now turn to the three conversion rules Aristotle established in Prior An¬
alytics A 2.
CR1-CR3 below are Aristotle’s statements of the three deduction rules involving
conversion. Our formulations of his statements using a modern notation exactly
reproduce Aristotle’s meaning, both in his manner of expression in Prior Ana¬
lytics and in his using them there. In the boxes below, the texts of Aristotle’s
models appear on the left, our modern representation of the pan valid argument
patterns on the right. 60 We treat Aristotle’s logical methodology for establishing
the conversion rules below (§3.1).
CR1 “It is necessary for a universal privative premiss of belonging to convert
with respect to its terms” (25a5~6).
“First, then, let premiss AB be universally privative”
e simple
(25al4):
conversion
“Now, if A belongs to none of the Bs, then neither will
1. AeB
B belong to any of the As.” (25al5-16)
.-. BeA
This can be expressed syntactically as: whenever AeB is taken then BeA can be
taken. Thus, a sentence fitting the pattern BeA logically follows from a given
sentence fitting the pattern AeB. Aristotle treats the e conversion first.
CR2 “And the attributive (xfjv xaxrjyopixrjv) premiss necessarily converts [with
respect to its terms], though not universally but in part” (25a7-9).
59 This rule, of course, applies to simple conversion, which likely was the beginning point for
ancient study of categorical sentence transformations. Here we witness Aristotle’s recording the
results of his study. Strictly, an a sentence does not convert, although he does not say as much.
Although he recognized that an o sentence is implied by its e counterpart, he did not provide an
e conversion per accidens. Aristotle noted that an o sentence does not convert, and he provided
a counterexample to establish this. Interestingly, since an e sentence converts simply, both an
o sentence and its converse are deducible; this, of course, does not establish that the one is a
logical consequence of the other.
60 We here have gathered from his text relating to the e conversion something of his thinking
about premiss formation and argument formation that our statements of rules below capture.
This manner of writing does not characterize the texts for the a and i conversions, which he
treats rather summarily; rather it is appropriately taken as given.
Aristotle’s Underlying Logic
169
On the universal attributive:
a conversion
per accidens
“And if A belongs to every B, then B will [necessarily]
1. AaB
belong to some A” (25al7-18)
.'. BiA
This can be expressed syntactically as: whenever AaB is taken then BiA can be
taken. Thus, a sentence fitting the pattern BiA logically follows from a given
sentence fitting the pattern AaB.
CR3 “The affirmative (xf)v xaxacpaxtxfjv) must convert partially [with respect
to its terms]” (25al0-ll).
On the partial attributive:
i simple
conversion
“And similarly if the premiss is partial: if A belongs to
1. AiB
some of the Bs, then necessarily B belongs to some of
.-. BiA
the As” (25a20-21)
This can be expressed syntactically as: whenever AiB is taken then BiA can be
taken. Thus, a sentence fitting the pattern BiA logically follows from a given sen¬
tence fitting the pattern AiB. Aristotle remarks that the partial privative sentence
does not convert: “... but the privative premiss need not [convert]” (25al2-13).
We might wish that Aristotle had stated each rule more rigorously. Nevertheless,
it is evident that he construes these conversions to be rules and that he treats them
syntactically. And, although he is quick to provide an example to help illustrate
his thinking in each case, 61 his doing so no more subverts his syntactic analysis and
configuration than does providing a counterargument subvert this for a modern
logician. Moreover, as we show below in relation to the syllogisms (§3.2), it is
evident from his discourse in A2 that Aristotle thinks of these transformations as
metalogical patterns of arguments whose premiss-sets are single sentences.
The two-premiss syllogism rules
Aristotle noted that two premisses with three different terms is the fewest number
by which someone could deduce a sentence that is neither (1) a repetition nor (2)
a conversion. Accordingly, to fulfill his principal concern in Prior Analytics, he
demonstrated which of these elementary two-premiss argument patterns have only
valid argument instances and which elementary patterns have only invalid argu¬
ment instances. The results of his study, particularly in Prior Analytics AJ,-7,
serve as elements, or principles — in particular, as deduction rules — in his de¬
duction system. Aristotle thought of syllogistic reasoning as progressively linking
61 We cite here Aristotle’s examples in the case of each rule. CR1: “For instance, if no pleasure
is a good, neither will any good be a pleasure” (25a6-7). CR2: “For instance, if every pleasure
is a good, then some good will be a pleasure” (25a9-10). CR3: “for if some pleasure is a good
then some good will be a pleasure” (25all-12).
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George Boger
the conclusions of two premiss arguments — to wit, the syllogisms — until a final
conclusion (theorem) is reached. Here again we extract his syntactic two-premiss
deduction rules before taking up his logical methodology (§3.2).
In order systematically to extract all possible panvalid patterns of two-premiss
categorical arguments, Aristotle considered in Prior Analytics A 4-6 every possible
arrangement of any two of the four kinds of categorical sentence with three different
terms. Working with a notion of ‘form’ of argument that is genuinely syntactic,
he systematically treated patterns of two protaseis (sentences), or premiss-pair
patterns, and their corresponding argument patterns, and he treated neither pre¬
misses nor arguments per se. To do this he treated each premiss-pair pattern
metalogically; that is, he treated at one time, say, all such patterns of two univer¬
sal attributive sentences in a given figure, by treating the one premiss-pair pattern
that they all fit in that figure. Aristotle modeled an object language syllogism
as an elementary two-premiss argument pattern — where two given sentences are
in the role of premises and one sentence is in the role of conclusion. In this way,
without considering object language arguments but only their patterns, he deter¬
mined which argument patterns are panvalid and which are not panvalid. Thus,
any argument fitting a panvalid pattern is valid; its validity might be recognized by
virtue of its fitting such a pattern. Aristotle determined that 14 such premise-pair
patterns are concludent and thus that at least one corresponding argument pattern
is panvalid and can thereby serve as a deduction rule. Aristotle first states each
of the syllogism rules in a sentence, as we would expect a rule to be expressed. He
then treats each of the argument patterns schematically, that is, he models each
as a two-premiss argument pattern to determine its panvalidity. Arguments are
introduced (1) to establish that certain premise-pair patterns are inconcludent and
(2) to serve as instances of panvalid argument patterns or of paninvalid argument
patterns.
We take Aristotle at his word when he states, on numerous occasions in both
Prior Analytics and Posterior Analytics , that “every demonstration [itacra ouioScl^lc]
and every deduction [udc; CTuXXoyiapoc;] will be through only three terms ... it will
also be from two premisses [or intervals] and no more, for three terms are two
premises” ( Pr. An. A25: 42a30-33; cf. Pr. An. A25: 41b36-37 and Po. An.
A19: 81bl0 & A25: 86b7-8 among many other passages). From this statement
and his practice throughout, we can extract his rules for syllogistic 62 premise-pair
formation as follows.
SPFR1 A syllogism premiss-set consists in two and only two of the four kinds of
categorical sentence.
Aristotle provides additional text in Prior Analytics A23 that confirms our taking
him to have such a rigid rule. We shall refer to this text again when we treat
62 We use ‘syllogistic’ in this section, and sometimes elsewhere, to refer to the syntax relations
among categorical sentences and not strictly to refer to consistent sets of categorical sentences
or valid categorical arguments. Cf. his definition of ‘sutlogismos’ in Pr. An. Al: 24bl8-20; see
below §5.1 n93.
Aristotle’s Underlying Logic
171
Aristotle’s notion of deducibility (§5.1).
Now, if someone should have to deduce A of B, either as belonging or
as not belonging, then it is necessary for him to take something about
something. If, then, A should be taken about B, then the initial thing
will have been taken. But if A should be taken about C, and C about
nothing nor anything else about it, nor some other thing about A, then
there will be no syllogism, for nothing results of necessity through a
single thing having been taken about one other. Consequently, another
premiss must be taken in addition. If, then, A is taken about something
else, or something else about it or about C, then nothing prevents
there being a syllogism, but it will not be in relation to B through the
premisses taken. Nor when C is taken to belong to something else, that
to another thing, and this to something else, but it is not connected
to B: there will not be a syllogism in relation to B in this way either.
(40b30-41a2)
Aristotle makes it abundantly clear that taking three terms in two sentences is
possible in only three ways. He established this implicitly in Prior Analytics A4-6,
but he makes this explicitly part of his argumentation in A 23 where he effectively
treats deducibility and the completeness of his logic. He writes there that “every
demonstration and every deduction must necessarily come about through the three
figures” (41bl-3; cf. 40bl9-22 & A28: 44b6-8, 19-20). His fuller statement at A23
follows:
For, in general, we said that there cannot ever be any syllogism of
one thing about another without some middle term having been taken
which is related in some way to each according to the kinds of pred¬
ications. For a syllogism, without qualification, is from premises; a
syllogism in relation to this term is from premisses in relation to this
term; and a syllogism of this term in relation to that is through pre¬
misses of this term in relation to that. And it is impossible to take
a premiss in relation to B without either predicating or rejecting any¬
thing of it, or again to get a syllogism of A in relation to B without
taking any common term, but <only> predicating or rejecting certain
things separately of each of them. As a result, something must be
taken as a middle term for both which will connect the predications,
since the syllogism will be of the term in relation to that. If, then, it
is necessary to take some common term in relation to both, and if this
is possible in three ways (for it is possible to do so by predicating A of
C and C of B, or by predicating C of both A and B, or by predicating
both A and B of C), and these ways are the figures stated, then it
is evident that every syllogism must come about through some one of
these figures. (41a2-18)
Thus, we can extract three additional syllogistic premiss formation rules.
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George Boger
SPFR2 The two categorical sentences in a syllogistic premiss-set consist in three
different non-logical constants (terms).
SPFR3 None of the three non-logical constants in a syllogistic premiss-set appears
twice in the same categorical sentence. Thus, the one categorical sentence
in a syllogistic premiss-set has a non-logical constant in common with the
other sentence in the premiss-set.
SPFR4 Any of the four logical constants may appear in each of the two categorical
sentences in a syllogistic premiss-set.
Aristotle nowhere states SPFR4 in rule fashion, but it is evident from his treatment
of the premise-pair patterns throughout Prior Analytics A4-6 that he consciously
works with such a rule of premiss formation.
The salient feature of his exposition in this connection is the crucial role he
attributes to the middle term. He makes this quite emphatic in Prior Analytics
A28:
It is also clear that one must take things which are the same, not
things which are different or contrary, as the terms selected for the
investigation. This is because, in the first place, the examination is for
the sake of the middle term, and one must take as middle something
the same, not something different. (44b38~45al; cf. A29: 45b36-46a2)
Again, there is a rule of the middle term for taking pairs of sentences as premisses
to form syllogistic arguments: “there cannot ever be any syllogism of one thing
about another without some middle term having been taken which is related in
some way to each according to the kinds of predications” ( A23\ 41a2-4). This,
of course, is a theme highly resonant in Posterior Analytics A & R, and this
might best be captured as follows from Posterior Analytics B4 : “A deduction
proves something of something through the middle term [6 pev yap ouAAoyiCTpoc;
xt xaxa tivdc; Seixvucu §ia xou peaou]” (91al4-15). Aristotle recognized three
possible positions for the term shared by each of the categorical sentences in a
syllogistic premise-set. He called this the middle term, and he named three figures,
“first”, “second”, and “third”. Accordingly, he had three rules for constructing
a syllogistic premise-set for each of the three figures (ayfjpaxa; singular oyfjpa).
While he names the first figure at the end of Prior Analytics A4 (26b33) Aristotle
defines it at the beginning. And thus we have his rule for forming a first figure
syllogistic premise-set.
SPFR5 I call the middle [term] which both is itself in another and has another
in it — this is also the middle in position — and call both that which is
itself in another and that which has another in it extremes [or extreme
terms]. (25b35-37)
Aristotle’s rule for forming a second figure syllogistic premiss-set is the following:
Aristotle’s Underlying Logic
173
SPFR6 When the same thing belongs to all of one term and to none of the other,
or to all of each or none of each, I call such a figure the second. In it,
I call that term the middle which is predicated of both and call those
of which this is predicated extremes; the major extreme is the one lying
next to the middle, while the minor extreme is the one farther from the
middle. The middle is placed outside the extremes and is first in position.
(26b34-39)
Aristotle’s rule for forming a third figure syllogistic premiss-set is the following:
SPFR7 If one term belongs to all and another to none of the same thing, or if
they both belong to all or none of it, I call such a figure the third. By
the middle in it I mean that term of which they are both predicated,
and by extremes the things predicated; by major extreme I mean the
one farther from the middle and by minor the one closer. The middle is
placed outside the extremes and is last in position. (28al0-15)
Thus, there are only three syntactic arrangements of middle (or common) terms,
called the three figures: “[1] by predicating A of C and C of B, or [2] by predicating
C of both A and B, or [3] by predicating both A and B of C” (41al5-16). In the
context of his logical investigations in Prior Analytics A4-6, we can state a rule
implicit in his treatment of the syllogisms.
SPFR8 Nothing is a syllogistic premiss-set except in virtue of these rules.
Thus far we have represented Aristotle’s rules for forming premise-sets of syl¬
logistic arguments. He also has syntax rules for forming syllogistic premise-
conclusion (P-c) arguments, that is, in particular, rules concerning the relation¬
ships of terms (1) to each other in the conclusion of a syllogistic argument in
relation to (2) those in the premise-set in connection with each figure. His rules
involve taking sentences to form P-c arguments consisting of a set of two sen¬
tences — call them protaseis (premises) — and a single sentence — call this the
CTupKEpaopa (sumperasma), or conclusion. These rules anticipate the rules of syl¬
logistic inference.
SAFR1 A two-premiss syllogistic argument consists in two and only two categor¬
ical sentences in the role of premisses and one and only one categorical
sentence in the role of conclusion.
SAFR2 Any of the four logical constants can appear in each of the three categor¬
ical sentences composing a syllogistic argument.
SAFR3 Each syllogistic argument consists in three and only three different non-
logical constants (terms); no non-logical constant appears twice in a cat¬
egorical sentence in a syllogistic argument.
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George Boger
SAFR4 Of the two different non-logical constants in the conclusion sentence, the
one appears once in one of the two premiss sentences, the other once in
the other premiss sentence.
SAFR5 In the first figure the predicate term of the conclusion is the term pred¬
icated of the middle term in the premiss-set; the subject term of the
conclusion is the term in the premiss-set of which the middle term is
predicated.
SAFR6 In the second figure the predicate term of the conclusion is the subject
term of the first or major premiss; the subject term in the conclusion is
the subject term of the second or minor premiss.
SAFR7 In the third figure the predicate term of the conclusion is the predicate
term of the first or major premiss; the subject term in the conclusion is
the predicate term of the second or minor premiss.
SAFR8 Nothing is a two-premiss syllogistic argument except in virtue of these
rules.
In each figure, the predicate term of the conclusion both is the major term and its
presence identifies the major premiss of a syllogistic argument; the subject term
of the conclusion both is the minor term and its presence identifies the minor
premiss of a syllogistic argument. These rules compass syllogistic P-c argument
formation for Aristotle in Prior Analytics. Thus, we can set out the patterns of
the three figures as follows, using our abbreviations of Aristotle’s logical constants
for convenience (Table 11. 63
Aristotle understood that the order of a given set of two categorical sentences
taken as premisses does not affect their implying a given categorical sentence taken
as a conclusion. Premiss order is not important for logical consequence. On occa¬
sion he reversed the order in which he presented the two sentence patterns in the
63 Explanation of Table 11. In Pr. An. Af-6 Aristotle established the syntax of the syllo¬
gisms or, more generally, of elementary syllogistic arguments. Throughout A4-6 he worked with
schematic letters for three terms in various premiss-pair patterns according to three figures. He
used, respectively, ABr, MNE, nPE (ABC, MNX, PRS). In each case, whether he stated first
the major or the minor premiss pattern, he always understood the predicate term (P) of the
conclusion pattern (PS or PxS) to be the first, or major, term — and to identify the major
premiss — in the premiss-pair pattern and the subject term (S) of the conclusion pattern to be
the last, or minor, term and to identify the minor premiss. We use ‘x’, ‘y\ and ‘.z’as placeholders
for any of the four logical constants. The term repeated in the premisses is the middle term
(M). This syntax is strict. Aristotle always considered the conclusion of an argument to fit the
sentence pattern PS (or PxS) and not its converse. (He did not, however, specifically show how
a given premiss-pair pattern does not result in a syllogism when the conclusion is converted, say,
in the case of Barbara, which we know to have no valid instances. But see Pr. An. B22.) In
this connection, when he treated Camestres and Disamis, he specifically converted the derived
sentence pattern to preserve this syntax. We have set out this syntax with our modern notation,
which exactly reproduces Aristotle’s meaning.
Aristotle’s Underlying Logic
175
Table 11.
Synopsis of Aristotle’s syllogistic argument patterns
First figure:
PMS: (ABT)
PxM, MyS|PzS
Second figure:
MPS: (MNE)
MxP, MyS|PzS
Third figure:
PSM: (nPE)
PxM, SyM|PzS
AB 1. PxM
BY 2. MyS
AT ? PzS
MN 1. MxP
ME 2. MyS
NE ? PzS
ns l. PxM
PE 2. SyM
nP ? PzS
premisses when he considered them in Prior Analytics A4-6. M This shows that he
understood this to be so. Nevertheless, in order to treat his premise-pair patterns
systematically, he treated them as ordered pairs in the framework of his strict
syllogistic syntax. Here the premiss order matters significantly (1) for systemati¬
cally treating all possible combinations (sc. patterns) of two categorical sentences
in the role of premisses and (2) for relating terms in the conclusion to those in
the premises. In general, he first treated the universal sentences as premises, then
combinations of universal and partial sentences, and finally combinations of partial
sentences.
Now, of the 192 possible combinations of syllogistic premises, Aristotle identified
14 that result in syllogisms when terms are substituted for the placeholders. 65
As he did with the one-premiss conversion rules, he did with the two-premiss
syllogistic rules. He first provided a sentence stating the rule before he represented
it schematically, and then he provided a metalogical proof of its panvalidity. Below
we provide Aristotle’s texts in Prior Analytics that model his 14 two-premiss
syllogism rules, our modern notation to the right. 66
SRI Whenever, then, three terms are so related to each other that the last is
in the middle as a whole and the middle is either in or not in the first
as a whole, it is necessary for there to be a complete syllogism of the
extremes. (25b32-35) laa
Barbara(25b37-39)
1. AoB
2. BaC
For if A is predicated of every B and B of every C, it is
.'. AaC
necessary for A to be predicated of every C.
64 This is the case, for example, when Aristotle treats third figure patterns. See Pr. An A6:
28a26-36, which includes Felapton, and A6\ 28b5-31, which treats Disarms, Datisis, and Bocardo.
For Aristotle, as for modern logicians, premiss order is independent of implication.
65 On there not being a fourth figure see below §3.5.
66 The ‘laa’, for example, after Aristotle’s statement of a rule, refers to the figure and the
patterns of each premiss sentence. In addition, rather than compose two statements for each of
two of Aristotle’s rule but which are expressed in one sentence, we have cited the passage twice,
as, for example, with the statement covering both Barbara and Celarent (SRI & SR2).
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George Boger
SR2 Whenever, then, three terms are so related to each other that the last is
in the middle as a whole and the middle is either in or not in the first
as a whole, it is necessary for there to be a complete syllogism of the
extremes. (25b32-35) lea
Celarent (25b40 - 26a2)
1. AeB
2. BaC
Similarly, if A is predicated of no B and B of every C,
.-. AeC
it is necessary that A will belong to no C.
SR3 If one of the terms is universal and the other is partial in relation to
the remaining term, then when the universal is put in relation to the
major extreme (whether this is positive or privative) and the partial is
put in relation to the minor extreme (which is positive), then there will
necessarily be a complete syllogism. (26al7-20) 1 at
Darii (26a23-25)
1. AaB
2. BiC
For let A belong to every B and B to some C. .
necessary for A to belong to some C.
.. it is
.'. A iC
SR4 If one of the terms is universal and the other is partial in relation to
the remaining term, then when the universal is put in relation to the
major extreme (whether this is positive or privative) and the partial is
put in relation to the minor extreme (which is positive), then there will
necessarily be a complete syllogism. (26al7-20) lei
Ferio (26a25-27)
1. AeB
2. BzC
And if A belongs to no B and B to some C, then it is
.-. AoC
necessary for A not to belong to some C.
SR5 When the terms are universal, there will be a syllogism when the middle
belongs to all of one term and none of the other, no matter which one
the privative is in relation to. (27a3-5) 2ea
Cesare (27a5-9)
1. MeN
2. MaX
For let M be predicated of no N but to every X. .
that N belongs to no X.
.. so
.-. NeX
SR6 When the terms are universal, there will be a syllogism when the middle
belongs to all of one term and none of the other, no matter which one
the privative is in relation to. (27a3-5) 2ae
Aristotle’s Underlying Logic
177
Camestres (27a9-14)
1. MaN
2. MeX
Next, if M belongs to every N but to no X, then neither
NeX
will N belong to any X.
SR7 If the middle is universal only in relation to one term, then when it is uni¬
versal in relation to the major extreme (whether positively or privatively)
but partially with respect to the minor and oppositely to the universal
... then it is necessary for the privative partial syllogism to come about.
(27a26-32) 2 ei
Festino (27a32-36)
1. MeN
2. MiX
For if M belongs to no N and to some X, it is necessary
.-. NoX
for N not to belong to some X.
SR8 If the middle is universal only in relation to one term, then when it is uni¬
versal in relation to the major extreme (whether positively or privatively)
but partially with respect to the minor and oppositely to the universal
... then it is necessary for the privative partial syllogism to come about.
(27a26-32) 2ao
Baroco (27a36-27bl)
1. MoN
2. MoX
Next, if M belongs to every N but does not belong to
.-. NoX
some X, it is necessary for N not to belong to some X.
SR9 For when both terms are positive [and universal], then there will be a
syllogism that one extreme belongs to some of the other extreme. (28a37-
39) 3 aa
Darapti (27al7-26)
1. PoS
2. RaS
When both P and R belong to every S, it results of
PiR
necessity that P will belong to some R.
SR10 And when one term is privative and the other affirmative [both universal],
then if the major term should be privative and the other term affirmative,
there will be a syllogism that one extreme does not belong to some of the
other. (28bl-3) 3ea
Felapton (28a26-30)
1. PeS
2. RaS
And if R belongs to every S but P to none, then there
.-. PoR
will be a deduction that P of necessity does not belong
to some R.
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George Boger
SR11 If one term is universal in relation to the middle and the other term is
partial then when both terms are positive it is necessary for a syllogism
to come about, no matter which of the terms is universal. (28b5-7) 3 ia
Disamis (28b7-ll)
1. PiS
2. RaS
For if R belongs to every S and P to some, then it is
PiR
necessary for P to belong to some R.
SR12 If one term is universal in relation to the middle and the other term is
partial then when both terms are positive it is necessary for a syllogism
to come about, no matter which of the terms is universal. (28b5-7) 3 ai
Datisis (28bll-15)
1. PaS
2. RiS
Next, if R belongs to some S and P to every S, then it
PiR
is necessary for P to belong to some R.
SR13 But if one term is positive, the other privative, and the positive term is
universal, then when the minor term is positive, there will be a syllogism.
(28bl5-17) 3 oa
Bocardo (28bl7-20)
1. PoS
2. RaS
For if R belongs to every S and P does not belong to
.. PoR
some, then it is necessary for P not to belong to some
R.
SR14 If the privative term is universal, then when the major term is privative
and the minor positive there will be a syllogism. (28b31-33) 3 ei
Ferison (28b33-35)
1. PeS
2. RiS
For if P belongs to no S and R belongs to some S, then
.'. PoR
P will not belong to some R.
Having now set out Aristotle’s deduction system, we turn to examine the logical,
or metalogical, methodology by which he established these deduction rules.
3 ARISTOTLE’S LOGICAL METHODOLOGY FOR ESTABLISHING
DEDUCTION RULES
In this section we examine the methods by which Aristotle established the rules of
his syllogistic deduction system. We also consider his methods for establishing that
Aristotle’s Underlying Logic
179
certain elementary argument patterns cannot serve as rules. To fulfill a principal
purpose in Prior Analytics, Aristotle demonstrated (1) which premiss patterns of
one sentence are concludent and which inconcludent when the terms of the given
sentence are converted in the conclusion. He also demonstrated (2) which premiss
patterns of two sentences are concludent and which inconcludent — that is, which
patterns when ‘interpreted’ result in a syllogism and which patterns do not result
in a syllogism. The results of his study, particularly in Prior Analytics A2, and
4-7, serve as rules in his deduction system.
To separate the elementary two-premiss argument patterns with only valid in¬
stances from the elementary two-premiss argument patterns with only invalid in¬
stances, Aristotle used two decision procedures in his metalogic: (1) the method
of completion , which he so named, and (2) the method of contrasted instances, 67
which itself has three modes. There are no elementary argument patterns with
both valid and invalid instances: none is neutrovalid. In the case of the two-premiss
patterns, Aristotle’s method is deductive, but not axiomatic, and enumerative. In
the case of the one-premiss patterns, his method is deductive but not a process of
completion per se.
Aristotle treated each conversion at Prior Analytics A2 in exactly the same
fashion as he treated the syllogisms at A4-6, namely, not as deductions per se
but as elementary argument patterns having only valid instances. He performs
metalogical deductions of the conversions at A2 just as he does for the syllogisms
at A5-6. 68 The points of similarity between his treatments of the one- and two-
premiss argument patterns show that when he examined the formal properties
of his logic, he treated the elementary panvalid argument patterns for syllogisms
and conversions as deduction rules. In this respect they are equally species of a
given genus. Moreover, Aristotle recognized there to exist valid arguments that are
instances of these patterns. Again, in respect of a given genus, these instances are
the same, namely, valid arguments. However, he never considered an instance of a
conversion to be an instance of a syllogism, and vice versa, just as a modern logician
would not consider an instance of a simplification to be an instance of an addition
in propositional logic. Below we first treat Aristotle’s logical methodology for
establishing the conversion rules (§3.1) and then his methodology for establishing
the syllogism rules (§3.2).
67 We follow W. D. Ross (1949: 302) in using the expression ‘contrasted instances’ to name
Aristotle’s method of establishing inconcludence.
68 Aristotle uses the same expressions in treating conversions and syllogisms in Pr. An.'.
‘xpoxacni;’ (premiss); ‘avayxrj’ (it is necessary that .. [25a6, 10, 21]) and oux dvdyxr) (it is
not necessary that ... [(25a23]); ‘avayxaiov’ (necessarily [25a8]) and oux avayxaiov (not neces¬
sarily [25al2]). Some corresponding examples from Pr. An. A4 to indicate this are: dvdyxr]
(25b38); avayxaiov (26a4); for the negation Aristotle writes “oux eaxai auXXoyiapoc;” (26a7-8).
He uses schematic letters (‘A’ and ‘B’) in exactly the same fashion. He equally treats the con¬
versions as rules of deduction for syllogistic reasoning as he does the syllogisms at A5-6. Of the
four possible conversions Aristotle treats each exactly, in principle, as he does when establishing
and reducing the two-premiss syllogism rules.
180
George Boger
3.1 Establishing the conversion rules
Aristotle treats converting the predicate and subject terms of each of the four kinds
of categorical sentence in Prior Analytics A2. After expressing each conversion
rule in a sentence (25a5-13), he then treats each schematically. He models each as
a one-premiss argument pattern and provides a deduction in the metalanguage of
Prior Analytics (25al4-26) to establish each conversion pattern to be a panvalid
argument pattern. This establishes its suitability as a rule. Then, in A 4-6 he
employs these conversions as rules in his metalogical deductions to establish the
panvalid patterns relating to the syllogisms.
Since there are only three conversions we treat each of them below. Aristotle
first treats establishing the panvalidity of the e conversion: 69
Now, if [el] A belongs to none of the Bs, then neither will B belong to
any of the As. For if it [B] does belong to some [A], for instance to C,
it will not be true that A belongs to none of the Bs, since C is one of
the Bs. [But it was taken to belong to none.] (25al5-17)
According to this text, then, we can represent Aristotle’s rather ‘intuitionist’ de¬
duction in the following familiar manner (Table 12). 70
Aristotle next provides a proof of the panvalidity of the a conversion per acci-
dens :
And if [el] A belongs to every B, then B will belong to some A. For if
it [B] belongs to none [A], neither will A belong to any B; but it [A]
was assumed to belong to every one [B]. (25al7-19)
Again, we can represent what he writes here in the following manner (Table 13).
On the proof of the panvalidity of the i conversion rule (Table 14):
If [el] A belongs to some of the Bs, then necessarily B belongs to some
of the As. (For if it [B] belongs to none [A], then neither will A belong
to any of the Bs [but it (A) was assumed to belong to some (B)].)
(25a20-22)
Finally, in a fashion less analogous to his methods of establishing inconcludence
of premise-pair patterns, Aristotle provides a counterexample to show that an 0
sentence does not convert (25a22-26; cf. 25al2-13). The result does not follow
necessarily (oux dvayxr] at 25a23 and oux dtvayxcuov at 25al2). His proof of the
paninvalidity of the 0 conversion follows:
69 Some logicians take this to be an instance of ekthesis early in Pr. An .; a later, more fully
presented instance appears in Pr. An. A6: 28a23-26.
70 We have aimed to represent something of Aristotle’s own thinking, although we have simpli¬
fied it here. Strictly, ‘C=A’ might not represent Aristotle’s thinking. Surely he would recognize
that some Cs might not be As, but the Cs Aristotle picks out are those of A to which B belongs,
as a Venn diagram might easily show. Cf. Smith, 1989, xxiii-xxv on ‘setting out’ and Smith,
1982 on ekthesis.
Aristotle’s Underlying Logic
181
Table 12.
Establishing the e conversion
rule
Aristotle’s text
Modern notation
1. A belongs to none of the
1. AeB
Bs
? neither will B belong to
? BeA
any of the As
2. it [B] does belong to some
2. BiA
assume
[A]
3. for instance [B belongs] to
3. C=A
basis of assumption
[every] C [to which A be-
longs]
4. C is one of the Bs
4. BiC
basis of assumption
5. it will not be true that A
5. -(AeB)
3,4 logic
belongs to none of the Bs
[or, that is, A will belong
[Az'B]
to some B]
6. [But it [A] was taken to
6. AeB & -(AeB)
1,5 conj & contra
belong to none [B].]
[AiB]
7. neither will B belong to
7. BeA
2-6 reductio
any of the As
Table 13.
Establishing the a conversion per accidens rule
Aristotle’s text
Modern notation
1. A belongs to every B
1. AaB
? B will belong to some A
? BiA
2. it [B] belongs to none [A]
2. BeA
assume
3. neither will A belong to
3. AeB
2 e-conversion
any B
4. it was assumed [A] to be-
4. AeB & AaB
3,1 conj & contra
long to every one [B]
5. [B will belong to some A]
5. BiA
2-4 reductio
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George Boger
Table 14.
Establishing the i conversion
rule
Aristotle’s text
Modern notation
1. A belongs to some of the
Bs
1. AiB
? B belongs to some of the
As
? BiA
2. it [B] belongs to none [A]
2. BeA
assume
3. neither will A belong to
any of the Bs
3. AeB
2 e-conversion
4. [it <A> was assumed] to
belong to some <B>]
4. AeB & AiB
3 conj & contra
5. [B belongs to some of the
As]
5. BiA
2-4 reductio
But if [ei] A does not belong to some B, it is not necessary for B also
not to belong to some A (for example if B is animal and A man: for
man does not belong to every animal, but animal belongs to every
man). (25a22-26)
“Man does not belong to some animal”, which is obviously true, does not convert
to “Animal does not belong to some man”, which is obviously false. We can set
this out in the following way (Table 15).
Table 15.
Establishing that converting an o sentence
does not produce a necessary result
Aristotle’s text
Aristotle’s
counterexample
1. A does not belong to some B
? B also does not belong to
some A
1. Man does not belong to some T
animal
? [Animal does not belong to F
every man.]
[But animal belongs to every T.
man.]
This proof uses an instance of the method of fact, which itself uses the principle
that no false sentence is implied by a true sentence. It is evident that Aristotle’s
notion of logical consequence is the same in each case and comports with that of
modern logicians (§5.3). His treatment of the conversion of categorical sentences
Aristotle’s Underlying Logic
183
was aimed to establish rules. The three one-premiss conversion rules can be rep¬
resented in the following schematic way with a modern notation that preserves
exactly what Aristotle accomplished in Prior Analytics A2 (Table 16):
Table 16.
Aristotle’s one-premiss conversion rules
a conversion
per accidens
e simple conversion
i simple conversion
1. AaB
1. AeB
1. AiB
BiA
.-. BeA
.-. BiA
3.2 Establishing the syllogism rules: deciding concludence of premiss-
pair patterns
Each syllogism fitting one of the four first figure panvalid patterns is teXeioc;
(teleios ), perfect or complete. This means that the necessity of its result following
from the things initially taken, its being valid, is obvious, or evident through itself,
to a participant (§5.1): nothing additional need be taken for this evidency (Al:
24bl8-24, AJf \ 26b28-33). While Aristotle did not prove 71 the panvalidity of the
teleioi sullogismoi patterns, he did think of the syllogisms fitting these patterns
as proving certain kinds of conclusion, that is, in particular, as each proving a
sentence in one of the four sentence patterns, or problemata: AaB, AeB, AzB,
and AoB. 72 And, moreover, while evidency of following necessarily is epistemic,
Aristotle understood a teleios sullogismos to be grounded in a corresponding ontic
reality that causes or makes this evidency possible, analogous to the truth-value
of a sentence, and thus also its panvalid pattern.
Each syllogism fitting a second and third figure pattern is aiEXrjt; ( ateles ), im¬
perfect or incomplete. The necessity of their conclusions is not obviously evident to
someone but Suvocxog, or potentially evident. Consequently, an epistemic process is
required to make the validity of an ateles sullogismos evident, namely, a deduction.
This distinction exactly characterizes the significant difference between Aristotle’s
treatment of second and third figure patterns from those of the first figure. Now,
just as an ateles sullogismos needs a deduction to establish its validity, so does its
71 W. D. Ross (1949: 22-28, 29), and other traditionalists such as J. N. Keynes (1906: 301)
and R. M. Eaton (1959: 86, 120), maintain that the first figure syllogisms are not primitive but
derived from the dictum de omni et nullo, which is variously conceived to be the “principle”
or the “axiom” of Aristotle’s system. On the other hand, J. Lukasiewicz (1958: 45, 46-47), J.
Corcoran (1974: 109), and J. Lear (1980: 2-3), for example, consider them to be given without
proof as self-evidently valid. Corcoran agrees with Lukasiewicz (1958: 46-47) to consider the
passage cited in Pr. An. (24b26-30; cf. Cat. Ibl0-12) that states the dictum de omni et nullo
to be definitional; cf. J. W. Miller (1938: 26-27).
72 He indicates this by writing, for example, of the first figure syllogisms that “all the problems
are proved by means of this figure [roxvxa xa Ttpo(3Xi)paxa Sctxvuxai 5ia xouxou xou axhpaxoq]”
(26b31), or by noting that the same results had been proved earlier (e.g., at 26b20-21).
184
George Boger
pattern need a metalogical deduction to establish its panvalidity. 73 In each case
of a second figure and a third figure panvalid pattern, Aristotle showed by means
of a metalogical deduction that a given premise-pair pattern is concludent, that is,
moreover, that such a premiss pattern results in an argument pattern such that an
arbitrary argument fitting this pattern is a syllogism. Aristotle used the method
of completion, TeXeioOaOai or TeXehoatt; ( teleiousthai or teleiosis), which here is
a deduction process carried out in the metalanguage of Prior Analytics , 74 Along
with the three conversion rules, this process explicitly employs the four panvalid
patterns of the first figure as deduction rules to establish which second and third
figure argument patterns are panvalid and could themselves, then, serve as rules.
Aristotle’s interest here was to establish which elementary argument patterns have
only valid argument instances. Every argument with semantically precise terms
fitting one of these patterns is valid. In this way he identified 14 panvalid patterns
in three figures. 75 Aristotle treated each pattern individually and not axiomat-
ically; his metasystematic treatment of the panvalid patterns involves induction
and is not strictly deductive. 76 What follows relates to the process of establish¬
ing, then, not the validity of object language arguments (sc. syllogisms), but the
panvalidity of their patterns corresponding to concludent premise-pair patterns in
the second and third figures.
The text concerning Camestres in the second figure illustrates that Aristotle
used a deduction in his metalanguage to establish that a given argument pattern
is in fact a panvalid pattern and not to establish that Camestres is itself a deduction
or itself derived from another ‘syllogism-axiom’.
If [el] M belongs to every N but to no X, then neither will N belong
to any X. For if M belongs to no X, neither does X belong to any M;
but M belonged to every N; therefore [apa], X will belong to no N (for
the first figure has again been generated [YEyevrftai yap xaXiv to itpanov
oyfipa]). And since the privative converts, neither will N belong to any
X. (27a9-14)
We can express exactly what Aristotle writes here in the manner of a deduction
with which we are familiar. Notice that Aristotle converts the conclusion to main¬
tain strict syllogistic syntax (Table 17).
73 See Pr. An. Al: 24b24-26; A5: 27al-3, 27al5-18, 28a4-7; A6: 28al5-17, 29al4-16.
74 The process of completion is, of course, also carried out in one or another object language.
75 Aristotle recognized 14 syllogisms in three figures, where traditionalist logicians, or
logicians referring to traditional logic, consider there to be 24 syllogisms in four figures. A good
recent reference for this information is William T. Parry k Edward A. Hacker 1991; an excellent,
older source is H. W. B. Joseph 1906.
76 The following analogy helps to explain Aristotle’s procedure. As a geologist might use a
hammer to break open a given rock to determine whether it is or is not a geode, and, upon
making the determination, place the object in one of two piles, so Aristotle used a metalogical
deduction to determine whether a given elementary argument pattern belongs in the set of
panvalid patterns and he used the method of contrasted instances to determine whether in the
set of paninvalid patterns.
Aristotle’s Underlying Logic
185
Table 17.
Establishing the panvalidity of a second
figure pattern: Camestres
Aristotle’s text
Modern notation
1. M belongs to every N
1. MaN
2. but [M] to no X
2. MeX
? neither will N belong to any X
? NeX
3. M belongs to no X
3. MeX 2 repetition
4. neither does X belong to any M
4. XeM 3 e-conversion
5. M belonged to every N
5. MaN 1 repetition
6. X will belong to no N
6. XeN 4,5 Clearent
7. neither will N belong to any X
7. NeX 6 e-conversion
The panvalidity of each second and each third figure pattern is determined in
just this manner, whether by direct (probative) or indirect ( reductio) deduction
using the conversion and teleioi sullogismoi rules. Table 18 indicates what first
figure pattern Aristotle used in the deduction process to establish the panvalidity
of patterns in the three figures.
Table 19 expresses in schematic notation exactly what Aristotle writes concern¬
ing each proof of the panvalidity of the patterns in Prior Analytics A5-6 (‘X’ =
contradiction).
It is important to understand that in Prior Analytics A5-6 Aristotle used his
syllogistic deduction system as part of his metalinguistic discourse to establish
certain features of the system itself. By this means he demonstrated 77 that a given
second or third figure argument pattern is panvalid since its conclusion pattern
follows logically from a given premise-pair pattern, which is thus understood to
be concludent. Proving that a given argument pattern is panvalid establishes
knowledge that the pattern can serve as a rule, since its extension is universal:
an arbitrary argument fitting this pattern is valid. 78 Establishing this is strictly
a metalogical process. Moreover, he demonstrated this by using the four patterns
of the teleioi sullogismoi ; this is indicated by his saying that “the first figure is
again generated” (27al2-13; cf. 27a8-9, 36). Aristotle intentionally used the first
figure patterns as deduction rules in A5-6, and he explicitly mentioned their use in
77 Aristotle’s “showings” are demonstrations as he remarks at 27b3 (on Baroco), 28a22-23 (on
Darapti), 28a28-29 (on Felapton), and at 28bl3-14 (on Datisi). His using the verb ‘SelxvuctSou’ at
27a8-9 (on Cesare), 27al4-15 (on Camestres and on Baroco), 28a29-30 (on Felapton), 28M4-15
(on Datisi), and at 28b20-21 (on Bocardo) confirms his intention.
78 Perhaps we see Aristotle here apply to the study of logic his requirement, expressed in Po.
An. A4, that something is proved universally of a kind when it is proved of an arbitrary instance
of that kind. He writes: “something holds universally when it is proved of an arbitrary and
primitive case. ...Thus, if an arbitrary primitive case is proved to have two right angles (or
whatever else), then it holds universally of this primitive item, and the demonstration applies to
it universally in itself’ (73b32-33, 73b39-74a2).
186
George Boger
Table 18.
on
Summary of Aristotle’s texts
completing each panvalid pattern
Pattern
Manner of proof: syllogism
Syllogism
considered
completed ...
used in the
completion
process
Barbara
by means of (Sid [dia]) the things
initially taken or through itself
(St aOxou)
(Barbara)
Celarent
dia the things initally taken or
through itself
(Celarent)
Darii
dia the things initally taken or
through itself
(Darii)
Ferio
dia the things initally taken or
through itself
(Ferio)
Cesare
probatively dia ...
Celarent
Camestres
probatively dia ...
Celarent
Festino
probatively dia...
Ferio
Baroco
by reducto ad impossible dia...
Barbara
Darapti
probatively dia. . .
Darii
Felapton
probatively dia. ..
Ferio
Disamis
probatively dia. ..
Darii
Datisi
probatively dia. . .
Darii
Bocardo
by reductio ad impossible dia...
Barbara
Ferison
probatively dia...
Ferio
this respect in his proofs concerning Cesare (27al2-13), Festino (27a36), Darapti
(28a22), and Ferison (28b34-35). Aristotle’s summary of A4-6 at A7 highlights
this point.
All the incomplete syllogisms are completed by means of
[TeXeLoOvTai Sid] the first figure. For they all come to a conclusion
[jiepchvovToci, i.e., are deduced] either probatively or through an absur¬
dity, and in both ways the first figure is generated
[ylvsToa]. (29a30-36; cf. A5 : 28a4-7 and A6: 29al4-16)
As he expressly stated he would, Aristotle determined in A4.-6 “how every syllo¬
gism is generated”. We complete Aristotle’s model of his logic by schematically
setting out his 14 syllogism rules using our abbreviations of Aristotle’s long form
to express the logical constants. Table 20 summarizes all the panvalid patterns
Aristotle’s Underlying Logic
187
Table 19.
Aristotle’s metalogical deductions
for each second and third figure panvalid pattern
Cesare (27a5-9)
Camestres (27a9 14)
Festino (27a32-36)
1. MeN
1. MaN
1. MeN
2. MaX
2. MeX
2. MiX
? NeX
? NeX
? NoX
3. NeM 1 e-con
3. XeM 2 e-con
3. Nem 1 e-con
4. MaX 2 rep
4. MaN 1 rep
4. MiX 2 rep
5. NeX 3,4 Celar
5 XeN 3,4 Celar
5. NoX 3,4 Ferio
6. NeX 5 e-con
Baroco (27a36-27b3)
1. MaN
2. MoX
3. NoX
3. NaX assume
4. MaN 1 rep
5. NaX 3 rep
6. MaX 4,5 Barb
7. MaX k
MoX 6,2 conj; X
8. NoX 3-7 reduct
Darapti (28al9-22)
Felapton (28a26-30)
Disamis (28b7-ll)
1. PaS
1. PeS
1. PiS
2. RaS
2. RaS
2. RaS
? PiR
? PoR
? PiR
3. PaS 1 rep
3. PeS 1 rep
3. SiP 1 i-con
4. SiR 2 a-con
4. SiR 2 a-con
4. RaS 2 rep
5. PiR 3,4 Darii
5. PoR 3,4 Ferio
5. SiP 3 rep
6. RiP 4,5 Darii
7. PiR 6 i-con
Datisi (28bll-13)
Bocarado (28bl7-20)
Ferison (28b33-35)
1. PaS
1. PoS
1. PeS
2. RiS
2. RaS
2. RiS
? PiR
? PoR
? PoR
3. SiR 2 i-con
3. PaR assume
3. PeS 1 rep
4. PaS 1 rep
4. RaS 2 rep
4. SiR 2 i-con
5. SiR 3 rep
5. PaS 3,4 Barb
5. PoR 3,4 Ferio
6. PiR 4,5 Darii
6. PaS
PoS 5,1 conj;X
7. PoR 3-6 reduct
188
George Boger
that might serve as rules in Aristotle’s deduction system.
Table 20.
Summary of Aristotle’s two-premiss syllogism
deduction rules using modern notation
First
Barbara
Celarent
Darii
Ferio
Figure
1. AaB
1. AeB
1. AaB
1. AeB
2. BaC
2. BaC
2. BiC
2. BiC
.-. AaC
.-. AeC
.'. AiC
.-. AoC
Second
Cesare
Camestres
Festino
Baroco
Figure
1. MeN
1. MaN
1. MeN
1. MaN
2. MaX
2. MeX
2. MiX
2. MaX
NeX
.-. NeX
.-. NoX
.'. NoX
Third
Darapti
Felapton
Disamis
Datisi
Bocardo
Ferison
Figure
1. PaS
1. PeS
1. PiS
1. PaS
1. PoS
1. PeS
2. RaS
2. RaS
2. RaS
2. RiS
2. RaS
2. RiS
.'. PiR
.-. PoR
.-. PiR
.-. PiR
.'. PoR
.'. PoR
When we treat Aristotle’s notion of deducibility (§5.1) we treat the process of
completion more fully than here. Nevertheless, we here briefly explain his meaning
of ‘ teleios ’. It is common to translate ‘ teleios ’ by ‘complete’, or even ‘perfect’, as
in ‘complete deduction’. However, taking a syllogism not to be a deduction but
a valid argument, we see that Aristotle would have taken a teleios sullogismos
to be an argument whose validity is obviously evident to a participant. Thus,
perhaps, a better translation of ‘ teleios sullogismos’, albeit a bit cumbersome but
more faithful to Aristotle’s meaning, would be ‘valid argument whose validity
is obviously apparent’; an ateles sullogismos would then be translated by ‘valid
argument whose validity is not obviously apparent’. Following this interpretation,
then, ‘ teleiousthai ’, which has been translated by ‘to be completed’, as in ‘all the
ateleis sullogismoi are completed by means of the first figure’, would mean ‘the
validity of those valid arguments whose validity is not apparent is made evident by
means of the first figure’. Aristotle used only the verbs ‘isXsiouaQca’ ( teleiousthai)
and ‘eTUieXeiaGai’ ( epiteleisthai ) in connection with using the patterns of the first
figure. Thus, in the deduction process, the validity of a. valid argument becomes
evident when during the process a teleios sullogismos appears, or ‘is generated’.
Such an appearance in a chain of reasoning signals to a participant the cogency of
the chain of inferences that links the conclusion sentence to the premiss sentences
as a logical consequence.
3.3 Establishing inconcludence of premiss-pair patterns
Aristotle used the method of contrasted instances in Prior Analytics AJ^-6 to
show that a given premiss-pair pattern is inconcludent, that it does not result
in a panvalid pattern — that no substitution of terms produces a syllogism, or
Aristotle’s Underlying Logic
189
valid argument. Consequently, at one stroke this method establishes the paninva¬
lidity of each of the four corresponding elementary argument patterns relating to
each premiss-pair pattern. Aristotle devised three processes for establishing the
inconcludence of a given premiss-pair pattern. Moreover, his method is signifi¬
cantly different than the method of counterargument. In this way he was able to
determine each case in which no syllogism is possible. His purpose was to elim¬
inate every elementary two-premiss argument pattern that could not serve as a
deduction rule. Below we treat each of the three related processes, establish the
conditions for concludence and inconcludence, and extract some semantic princi¬
ples underlying Aristotle’s methods (§3.4). We also address two possible objections
to interpreting Aristotle as treating premiss-pair patterns (§3.5.).
The method of contrasted instances
Aristotle introduced his most commonly used method, the method of contrasted
instances, for deciding inconcludence at Prior Analytics Af and used it throughout
Af-6. He writes, in relation to the premiss-pair pattern lae:
However, if [ei]the first extreme follows [i.e., belongs to] all the middle
and the middle belongs to none of the last, there will not be a syllogism
of the extremes, for nothing necessarily results in virtue of these things
being so. (26a2-5)
This sentence states a set of formal relationships of three terms in two universal
sentences in the role of premisses for not generating a syllogism in the first figure.
This passage states the conditions concerning the pattern PaM, MeS | for which
no categorical sentence is a logical consequence of two other categorical sentences
fitting this pattern. Thus, Aristotle eliminates four elementary argument patterns
in the standard syntax as possible syllogisms by establishing their premiss pattern
to be inconcludent. He continues:
For it is possible [evSexetoci] f or the first extreme to belong to all as well
as to none of the last. Consequently, neither a partial nor a universal
conclusion results necessarily [ylvexat dvayxchov]; and, since nothing
is necessary because of these, there will not be a syllogism. Terms
for belonging to every are animal, man, horse; for belonging to none,
animal, man, stone. (26a5-9)
We can express what he writes here as follows (truth-values to the right) (Table
21 ) :
For Aristotle this demonstrates 79 that “nothing necessarily results”, that no
valid argument (syllogism) is possible from sentences fitting this premiss-pair pat-
79 He considers himself to demonstrate the inconcludence of certain premiss-pair patterns as
is indicated by his writing, for example, that some results must be proved (Seixteov) in another
way (see Pr. An. A5: 27b20-21, 28; A6: 29a6).
190
George Boger
Table 21.
Establishing inconcludence by
the method of contrasted instances
Pattern: PaM, MeS — PaS
1
Animal [A] belongs to every man [M],
AaM
T
2
Man belongs to no horse [H].
MeH
T
?
Animal belongs to every horse.
AaH
T
PattermPaM, MeS|PeS
1 .
Animal [A] belongs to every man [M],
AaM
T
2.
Man belongs to no stone [S].
MeS
T
?
Animal belongs to no stone.
AeS
T
tern since, as he shows, the results “could be otherwise". Aristotle clearly uses nei¬
ther the method of counterargument nor the method of counterinterpretation, 80
each of which requires finding an instance of an argument having true premisses
and a false conclusion in the same form as a given argument. Rather, by substitut¬
ing two sets of three terms for the schematic letters, he constructs two arguments
each of whose premisses are known to be true sentences fitting the same premiss-
pair pattern and whose conclusions also are known to be true sentences, but in
the one argument it is an a sentence, in the other an e sentence. It is not possible
to do this with a concludent pair, since every similar substitution that produces
true sentences as premisses will result in at least one false sentence, either the
a or the e sentence, in the conclusion. Thus, any two sentences of three terms
fitting this premiss-pair pattern are shown never to result together in a valid ar¬
gument. This premiss-pair pattern is inconcludent. No syllogism is possible in
this case. It is evident, moreover, that Aristotle treats at one time in this way
four argument patterns in the standard syntax for each premiss-pair pattern; he
does not show that each of the four patterns is paninvalid by using counterargu¬
ments. With 26a5-9 Aristotle establishes a practice that he uses throughout A4-6
to demonstrate inconcludence. This method of deciding inconcludence works for
almost every premiss-pair pattern, noticeably failing in some instances when the
minor premiss is a partial, and usually a privative, sentence.
This method of deciding inconcludence, while different than, is nevertheless
easily adapted to the method of counterargument, but adapted at the metalogical
level. Both methods achieve the same results. We can apply Aristotle’s two sets
of three terms to the two argument patterns but switch the terms for belonging to
none to belonging to every and vice versa, and then make the substitutions in the
argument patterns accordingly. Thus (Table 22):
80 A counterargument is an argument in the same form as a given argument (whose invalidity
is to be established) but has premisses that are true and a conclusion that is false. A counter¬
interpretation is an argument in the same form as a given argument (whose invalidity is to be
established) but a model of the premiss-set is not a model of the conclusion.
Aristotle’s Underlying Logic
191
Table 22.
Method of counterargument
for establishing invalidity
Pattern; PoM, MeS|PoS
1. Animal [A] belongs to every man [M],
AoM
T
2. Man e stone [S].
MeS
T
? Animal belongs to every stone.
AaS
F
? Animal belongs to some stone
AiS
F
Pattern: PaM, MeS|PeS
1. Animal [A] belongs to every man [M].
AaM
T
2. Man belongs to no horse [H].
MeH
T
? Animal belongs to no horse.
AeH
F
? Animal does not belong to some horse.
AoH
F
In these cases each sentence of the premiss-set is true and the respective conclu¬
sion sentences are false. Here, then, are counterarguments for the arguments pro¬
vided by Aristotle, which may serve as modern counterparts to Aristotle’s ancient
method. The method of counterargument, could also establish that a given pattern
is paninvalid, insofar as no syllogistic pattern is neutrovalid, on the principle that
two arguments in the same pattern are either both valid or both invalid.
It is apparent that Aristotle did not use the method of counterargument in
Af-6. Moreover, Aristotle did not establish arguments to be invalid but argument
patterns to be paninvalid, and, more specifically, he established the inconcludence
of certain premiss patterns of two categorical sentences with places for three dif¬
ferent terms in his search for syllogistic rules. He knew this procedure to establish
the paninvalidity at one time of four elementary argument patterns.
Conditions of concludence and inconcludence
Aristotle did not invent a name for his method of deciding inconcludence, nor
did he invent expressions that denote features or principles of his method. Nev¬
ertheless, it is apparent that he consciously worked with a notion of contrariety
that pertains to two categorical sentences each of which is the conclusion of a
different categorical argument whose premiss sentences fit the same premiss-pair
pattern. Here we describe two conditions 81 that underlie Aristotle’s two decision
procedures, the one pertaining to concludent patterns the other to inconcludent
patterns.
In the context of Prior Analytics Af-6 “to be otherwise” or “it is possible to
be otherwise” involves a notion of contrariety according to which it is logically
81 Concerning these two conditions, the expression ‘condition I’ is an abbreviation for the
four points under it; likewise for ‘condition II’. This simplifies treating the semantic principles
underlying Aristotle’s thinking treated in §3.4.
192
George Boger
Condition I: pertaining to concludent patterns resulting in a
syllogism
Of two sentences:
1. Each sentence is the conclusion of a categorical argument, each argument
has the same premiss-pair pattern into which are substutited different sets
of three terms that produce true sentences.
2. Each sentnece has a different set of predicate and subject terms
3. Each sentence is a universal categorical sentence: the one at an a sentence,
the other an e sentence
4. It is logically impossible for both sentences to be true.
Condition II: pertaining to inconcludent patterns not resulting
in a syllogism
Of two sentences:
1. Each sentence is the conclusion of a categorical argument, each argument
has the same premiss-pair pattern into which are substutited different sets
of three terms that produce true sentences.
2. Each sentence has a different set of predicate and subject terms
3. Each sentence is a universal categorical sentence: the one an a sentence,
the other an e sentence
4. It is logically possible for both sentences to be true.
possible for substitution instances for a given premiss-pair pattern to produce
conclusion sentences that satisfy condition II. Were a given premiss-pair pattern
concludent, it would be logically impossible for an arbitrary substitution instance
to produce conclusion sentences not satisfying condition I.
The modified method of contrasted instances
Aristotle uses a modified method of contrasted instances for deciding inconcludence
in Prior Analytics Af for treating only two premiss-pair patterns, PaM, MoS |
and PeM, MoS | (26a39-bl4), both of which are then immediately treated by
his third (or second most used) method, that of deducing inconcludence from the
indeterminate. We cite 26a39-bl4 in its entirety to examine his modified method.
Nor will there be a syllogism whenever [oxav] the term in relation to
the major extreme is universally either attributive or privative, and the
term in relation to the minor is partially privative_ (26a39-26b3)
This sentence refers to two sets of necessary relationships of three terms in two
premiss sentences, the one universal, whether attributive or privative, the other
partially privative, for not generating a syllogism in the first figure. The sentence in
the minor premiss in each case is an o sentence. This passage states the conditions
under which substituting sets of three terms into the premiss-pair patterns PaM,
Aristotle’s Underlying Logic
193
MoS | and PeM, MoS |, covering eight argument patterns in the standard syntax,
never results in a syllogism, and, accordingly, it asserts the paninvalidity of these
eight argument patterns. Here, however, Aristotle noticeably uses a variant of his
most commonly used method for deciding inconcludence. The passage continues:
... as, for instance, if A belongs to every B and B does not belong to
some C, or if it does not belong to every C. For whatever part <of the
last extreme> it may be that the middle does not belong to, the first
extreme could follow [&xoXou0f)aei] all as well as none of this part. For
let the terms animal, man, and white be assumed, and next let swan
and snow also be selected from among those white things of which man
is not predicated. Then, animal is predicated of all of one but of none
of the other, so that there will not be a syllogism. (26b3-10)
We can express what he writes here concerning PaM, MoS | as we did above for
PaM, MeS | (Table 23).
Table 23.
Establishing inconcludence by
the modified method of contrasted instances
Pattern: PaM, MoS|PaS
1. Animal [A] belongs to every man [M]
2. Man does not belong to some
white [selecting swan] [W/S]
AaM
MoW/S
T
T
? Animal belongs to every swan
AaW/S
T
Pattern: PaM, MoS|PeS
1. Animal [A] belongs to every man [M].
2. Man does not belong to some white
[selecting snow] [W/S].
AaM
BoW/S
T
T
? Animal belongs to no snow
AeW/S
T
The difference here consists in his “selecting from among white things” to which
“man” does not belong. Aristotle here takes his substances to exist, that they
are members of non-empty classes. Moreover, he takes ‘some’ in its determinate
sense. Still, condition II is satisfied for these conclusion sentences as in the case
for PaM, MeS | and the others. Aristotle next turns to PeM, MoS | in the same
fashion. He writes:
Next, let A belong to no B and B not belong to some C, and let the
terms be inanimate, man, white. Then, let swan and snow be selected
from among those white things of which man is not predicated (for
inanimate is predicated of all of one and of none of the other). (26bl0-
14)
194
George Boger
Table 24.
Establishing inconcludence by the
modified method of contrasted instances
Pattern: PaM, MoS|PaS
1. Inanimate [I] belongs to no man [M]
2. Man does not belong to some
white [selecting snow] [W/S]
IeM
MoW/S
T
T
? Inanimate belongs to every snow
IaW/S
T
Pattern: PeM, MoS|PeS
1. Inanimate [I] belongs to no man [M].
2. Man does not belong to some
white [selecting swan] [W/S].
IeM
MoW/S
T
T
? Inanimate belongs to no swan
IeW/S
T
We can set this out as follows (Table 24).
This is similar to his most commonly used method of deciding inconcludence, and
we can easily grasp Aristotle’s application. The conclusion sentences again satisfy
condition II.
The method of deducing inconcludence from the indeterminate
Finally, Aristotle uses a method of establishing that a given premiss-pair pattern
is inconcludent by deciding its inconcludence from another pattern already deter¬
mined to be inconcludent. This method is introduced at 26bl4-21 in relation to
the premiss-pair patterns PaM, MoS | and PeM, MoS |. Two matters are note¬
worthy: (1) Aristotle distinguishes an indeterminate (aStopicrcoc;) sentence from a
(determinate) partial (pepoc;) sentence; and (2) he alludes to his having shown the
pattern PaM, MeS | (at 26a2-9) not to generate a syllogism. His method may be
designated, as he himself virtually does, as the method of deducing inconcludence
from the indeterminate. We cite the passage from Af, where he first introduces
this method, but we examine the method more fully below in connection with
another passage from Prior Analytics A5.
Moreover, since “B does not belong to some C” is indeterminate, that
is, it is true if B belongs to none as well as if it does not belong to every
(because it does not belong to some), and since a syllogism does not
come about when terms are taken such that B belongs to none (this
was said earlier [at 26a2-9]), then it is evident that there will not be a
syllogism on account of the terms being in this relationship either (for
there would also be one in the case of these terms). It may <also> be
proved similarly if the universal is put as privative. (26bl4-21)
Aristotle’s Underlying Logic
195
Since (1) BoC is indeterminate, BoC is true if BeC is true, and (2) PaM, MeS |
has already been shown to result in nothing necessarily, then PaM, MoS | cannot
result in anything necessarily. PaM, MoS | is a weaker form of PaM, MeS |, which
had been shown to be inconcludent. When BoC is partial and indeterminate, that
B does not belong to some Cs and possibly to no Cs, then, since nothing has
been specified about those Cs that are B, we are lead back to the aspect just
treated. In other words, it is possible either (1) that a substitution instance could
result in sentences fitting AaC and AeC both being true or (2) that a substitution
instance produces true premiss sentences and a false conclusion sentence. The
sets of substitution terms in the premiss-pair pattern PaM, MoS | generate no
necessary result. The proof is the same for the pattern PeM, MoS |.
This method of deciding inconcludence is perhaps better portrayed in the fol¬
lowing passage from A 5 where Aristotle treats premiss-pair patterns of the second
figure. Again, we cite the passage in its entirety.
But whenever[dTav] the premisses are the same in form (that is, both
are privative or both attributive), then in no way will there be a syllo¬
gism. (27bl0-12)
This sentence refers to four sets of necessary relationships of three terms in two
premiss sentences, the one universal, the other partial, whether minor or major,
and both sentences having the same quality, for not generating a syllogism in the
second figure. This passage states conditions of inconcludence covering the four
premiss-pair patterns MeP, MoS—, MaP, MiS—, MoP, MeS—, and MiP, MaS—.
Accordingly, it establishes the paninvalidity of sixteen argument patterns in the
standard syntax. The passage continues:
For let the premisses first be privative, and let the universal be put in
relation to the major extreme (that is, let M belong to no N and not
to some X). It is then possible [evSexexca] for N to belong to every X
as well as to none. Terms for not belonging are black, snow, animal.
(27bl2-16)
Here Aristotle shows that the premiss-pair pattern MeP, MoS— is inconcludent.
The usual practice would be to produce two arguments whose premiss sentences fit
the same premiss-pair pattern and whose conclusion sentences satisfy condition II.
He is able to get terms for belonging to none , but he cannot get terms for belonging
to every to satisfy condition II. We can illustrate what he writes here as follows
(Table 25):
This is all familiar. Aristotle then continues to treat the argument pattern MeP,
MoS | PaS using the method of deducing from the indeterminate. He writes:
We cannot get [oux ecru Xapeiv] terms for belonging if M belongs to
some X and does not belong to some (for if N belongs to every X and
M to no N, then M will belong to no X: but it was assumed to belong
196
George Boger
Table 25.
Establishing inconcludence by the
method of proving from the indeterminate
Pattern: PaM, MoS|PaS
1.
2.
MeN
MoX
?
MoX
Pattern: MeP, MoS|PeS
1. Black [B] belongs to no snow [Sj.
2. Black does not belong to some animal
white [selecting swan] [W/S].
BeS
Bo A
T
T
? Snow belongs to no animal
SeA
T
to some). It is not possible to get [oux EyxwpeL Xa(3ei.v] terms in this
way, then, but it must be proved from the indeterminate [ex Se xou
aStopiaxou Sexxeov], For since “M does not belong to some X” is also
true even if M belongs to no X and there was not a syllogism when it
belonged to none [27a20-25], then it is evident that there will not be
one in the present case either. (27bl6--23)
Aristotle does not use his modified method of deciding inconcludence in this case
because he could not get terms for belonging to every for the argument pattern
MeP, MoS | PaS when the minor premiss is an o sentence. Every set of term
substitutions exhibiting true premiss sentences results in producing a false con¬
clusion sentence — a situation that could, in fact, establish the paninvalidity of
this argument pattern by the method of fact. However, he aims to establish the
inconcludence of premiss-pair patterns and thereby the paninvalidity of their cor¬
responding argument patterns. Accordingly, he takes the o sentence to be indeter¬
minate and refers back to his determining, by the method of contrasted instances,
that the premiss-pair pattern MeP, MeS | does not generate a syllogism. Since
(1) the minor premiss sentence in MeP, MoS | is a weaker privative of the minor
premiss sentence in MeP, MeS |, and (2) the truth-value of a partial follows its
corresponding universal (when true), then, since (3) MeP, MeS | is a stronger
premiss-pair pattern than MeP, MoS | and (4) MeP, MeS | has been shown to be
an inconcludent pattern, (5) what is true of MeP, MeS | is also true of MeP, MoS
|; thus, MeP, MoS | is an inconcludent premiss-pair pattern. Aristotle’s reasoning
can be expressed as follows:
1. MeP, MeS | Premiss-pair pattern known to
be inconcludent
2. MeP, MoS| e premiss weakening:
Premiss-pair pattern to be estab¬
lished as inconcludent
Aristotle’s Underlying Logic
197
Again, he takes his terms here to relate to non-empty classes. This method,
of course, depends for its success upon a prior use of the method of contrasted
instances for deciding inconcludence. Tables 26-28 below catalogue each instance
of Aristotle’s deciding inconcludence. 82
3-4 Semantic principles underlying Aristotle’s method of deciding in¬
validity
Aristotle considered the following expressions to be synonymous: “the results could
be otherwise” or “it is [logically] possible for the results to be otherwise”; “nothing
results necessarily”; “nothing follows necessarily”; “the results are not [logically]
necessary”; and, in the case of the syllogisms, “there is not a syllogism of the
extremes”. These expressions, of course, relate to invalidity and paninvalidity and
have their counterparts for validity and pan validity. And, moreover, in relation to
our topic, they all involve a notion of contrariety special to categorical sentences
in the role of conclusions in categorical arguments.
Aristotle recognized and worked with syntactic and semantic principles by which
he established that a given premiss-pair pattern is concludent and another in-
concludent. This pertains to satisfying condition I and condition II respectively.
Adapting Aristotle’s method of establishing invalidity to the method of counterar¬
gument indicates the correctness of his method. However, our doing this does not
provide insight into the principles underlying his method, which principally aims
at premiss-pair patterns and consequently their corresponding argument patterns.
These principles were unexpressed by Aristotle as others have been expressed by
modern logicians respecting the method of counterargument. All the following
principles pertain to premiss-pair patterns and to their corresponding categorical
argument patterns and arguments. We believe these principles represent Aristo¬
tle’s own thinking about deciding invalidity. Moreover, they provide an interesting
insight into an ancient logic that might shed light on modern logics.
82 In Tables 26-28, ‘PP’ refers to the various premiss-pair patterns Aristotle treats, and ‘SL’
refers to his providing schematic letters in treating a given pattern. Aristotle sets out terms
according to the schematic order for each figure’s schematic letters: first figure-PMS, second-
MPS, third-PSM.
PP Treatment
ae 26a5-9
ee 26all-13
ia 26a33-36
oa 26a33-36
ie 26a36-39
oe 26a36-39
ao 26b3-10
26bl4-20
eo 26bl0-14
26b20-21
Table 26.
Covering
Statement
26a2-4
26a9-ll
26a30-33
26a30-33
26a30-33
26a30-33
26a39-26b3
26a39-26b3
Catalogue of premiss-pair patterns not generating a syllogism in the first figure (Ai)
ISL I Method of Establishing Inconcludence
Contrasted instances
Modified contrasted instances
Deduction from
the Indeterminate
Terms for belonging
_ PMS _
no animal-man-horse
no science-line-medicine
yes good-condition-wisdom
yes good-condition-wisdom
yes white-horse-swan
yes white-horse-swan
Terms for not belonging
_ PMS _
animal-man-stone
science-line-unit
good-condition-ignorance
good-condition-ignorance
white-horse-raven
white-horse-raven
Terms for belonging
Terms for not
belonging
animal-man-white animal-man-white From loe
/swan (26b3-10)_/snow_26M4-20
inanimate-man-white ammal-man-white From lee
/snow (26bl0-14) /swan 26b(14-20)20-21
198 George Boger
Table 27. Catalogue of premiss-pair patterns not generating a syllogism in the second figure (A5)
pp
Treatment
Covering
Statement
SL
Method of Establishing Inconcludence
Contrasted instances
Modified contrasted instances
Deduction from
the Indeterminate
Terms for belonging
MPS
Terms for not belonging
MPS
Terms for belonging
Terms for not
belonging
aa
27al8-20
27a23-25
yes
substance-animal-man
substance-animal-number
ee
27a20-23
27a23-25
yes
line-animal-man
line-animal-stone
oa
27b4-6
27b9-10
yes
animal-man-raven
animal-white-raven
ie
27b6-8
27b9-10
yes
animal-substance-unit
animal-substance-science
eo
27bl2-23
27bl0-12
(27b34-36)
yes
black-snow-animal
From 2ee
27bl6-23
ai
27b23“28
27bl0-12
(27b34-36)
yes
white-swan-stone
oe
27b28-32
27bl0-12
(27b34-36)
yes
white-animal-raven
white-stone-raven
-—
la
27b32-34
27bl0-12
(27b34-36)
no
white-animal-swan
white-animal-snow
ii
27b38-39
27b36-38
no
white-animal-man
white-animal-inanimate
00
27b38-39
27b36—38
no
white-animal-man
white-animal-inanimate
10
27b38-39
27b36-38
no
wite-animal-man
white-animal-inanimate
01
27b38-39
27b36-38
no
white-animal-man
white-animal-inanimate
Aristotle’s Underlying Logic 199
Table 28. Catalogue of premiss-pair patterns not generating a syllogism in the third figure (>16)
pp
Treatment
Covering
Statement
SL
Method of Establishing Inconcludence
Contrasted instances
Modified contrasted instances
Deduction from
the Indeterminate
Terms for belonging
PSM
Terms for not belonging
PSM
Terms for belonging
Terms for not
belonging
ae
28a30-33
(28bl-3)
28b3-4
yes
animal-horse-man
an i mal- i nan i mate- man
ee
28a33-36
(28a37-39)
28a39-28bl
no
animal-horse-inanimate
man-horse-inanimaate
ao
28b22-31
28b22-23
yes
animate-man-animal
From 3 ae
28b24-31
ie
28b36-38
28b36
no
animal-man-wild
animal-science-wild
oe
28b39-29a2
28b38-39
no
animal-science-wild
animal-man-wild
eo
29a2-6
28b38-39
yes
raven-snow-white
From 3ee
29a3-6
ii
29a9-10
29a6-9
animal-man-white
animal-inanimate-white
00
29a9-10
29a6-9
animal-man-white
animal-inanimate-white
io
29a9-10
no
animal-man-white
animal-inanimate-white
oi
29a9-10
animal-man-white
animal-inanimate-white
200 George Boger
Aristotle’s Underlying Logic
201
1. A given premiss-pair pattern is concludent if and only if every set of term
substitutions satisfies condition I.
2. A given premiss-pair pattern is inconcludent if and only if no set of term
substitutions satisfies condition I (or every set of term substitutions satisfies
condition II).
3. Two arguments having sentences that fit the same premiss-pair pattern and
sentences whose conclusions satisfy condition I cannot both be valid (but, of
course, both may be invalid as with Darii).
4. Two arguments having sentences that fit the same premiss-pair pattern and
sentences whose conclusions satisfy condition II are both invalid.
5. No argument having all true premisses and a false conclusion is valid.
6. A given argument pattern is panvalid if and only if it is logically impossible
for an arbitrary argument fitting the given pattern to have true premisses
and a false conclusion.
7. A given argument pattern is paninvalid if and only if it is logically impossible
for an arbitrary argument fitting the given pattern to be valid.
8. A given argument is valid if it fits a panvalid argument pattern. The sen¬
tence in the conclusion follows necessarily (it cannot be otherwise) from the
sentences comprising the premiss-set.
9. A given argument is invalid if it fits a paninvalid argument pattern. No sen¬
tence follows necessarily (the results can be otherwise) from other sentences
in a premiss-set.
Of course, in respect of numbers 8 and 9, a pattern does not make a given argument
valid or invalid. Rather an argument is valid just in case all the information
contained in the conclusion sentence is already contained in the premiss sentences,
invalid if more information is in the conclusion than in the premisss. Again, just
as Aristotle believes that truth follows being, so does he believe that validity,
or following necessarily, follows being: there is an ontic underpinning for a valid
argument’s validity just as there is for a true sentence’s truth.
Some epistemic principles relating to the semantic principles listed above include
the following:
1. It is sufficient for knowledge of the concludence of a given premiss-pair
pattern to produce two arguments whose conclusion sentences satisfy condi¬
tion I.
2. It is sufficient for knowledge of the inconcludence of a given premiss-pair
pattern to produce two arguments whose conclusion sentences satisfy condi¬
tion II.
202
George Boger
In some cases, when the minor premiss is an indeterminate sentence, it is
sufficient to demonstrate that the given pattern is a weaker form of a premiss-
pair pattern already established to be inconcludent.
3. It is sufficient for knowledge of the panvalidity of a given argument pattern
to produce either a direct or reductio deduction of its conclusion sentence
pattern.
4. It is sufficient for knowledge of the paninvalidity of a given argument pattern
to show that its premiss-pair pattern is inconcludent, or in some cases (§3.5)
to show that a given argument pattern fits an argument pattern rejected in
the case of a concludent premiss-pair pattern.
5. It is sufficient for knowledge of the validity of a given argument to show that
it fits a panvalid argument pattern.
6. It is sufficient for knowledge of the invalidity of a given argument to show
that it fits a paninvalid argument pattern, or to show that the sentences of
its premiss-set fit an inconcludent premiss-pair pattern.
3.5 Determining the paninvalid argument patterns of concludent pre¬
miss patterns
Two problems seem to arise were it true that Aristotle exclusively treated patterns
of premisses and argument patterns in Prior Analytics Af-6 rather than directly
treating arguments. Consider the following two arguments, A1 and A2, both of
which are invalid, however, both of which also have a premiss pattern established
to be concludent (i.e., relating to Barbara).
Table 29.
A1 Pattern: PaM, MaS|PeS
1. Animal [A] belongs to every mammal [M].
AaM
T
2. Mammal belongs to every human [H].
Mgll
T
? Animal belongs to no human
AeH
F
A2 Pattern: PaM, MaS SaP
1. Animal [A] belongs to every mammal [M]
AaM
T
2. Mammal belongs to every human [H].
MaH
T
? Human belongs to every animal.
HaA
F
These two arguments are obviously invalid on the principle that no argument is
valid having true premisses and a false conclusion. We cite two other arguments
that perhaps better illustrate the problems because their invalidity may not be
immediately evident. Note that each premiss-pair pattern is concludent: Darii
results from that in A3, Baroco in A4.
Aristotle’s Underlying Logic
203
Table 30.
A3 Pattern: PaM, MfSjPaS
1. Surface [S] belongs to every table [T].
SaT
T
2. Table belongs to some furniture [F],
TiF
T
? Surface belongs to every furniture.
SaF
F
A4 Pattern: MaP, MoS|SoP
1. Container [C] belongs to every bottle [B]
CaB
T
2. Container belongs not to every plastic [P].
CoP
T
? Plastic belongs not to every bottle.
PoB
F
These arguments represent the two basic concerns that a modern logician might
have. (1) Since not every argument relating to a concludent pattern is valid, how
are the paninvalid argument patterns of a given concludent premiss-pair pattern
identified when the given syllogistic syntax is standard? (2) Similarly, how are the
paninvalid argument patterns of a given concludent premiss-pair pattern deter¬
mined when the syllogistic syntax is converted in the conclusion? These questions
pertain to every paninvalid pattern associated with a concludent premiss-pair pat¬
tern in any of the three figures, whether the pattern’s conclusion is a universal
or a partial sentence. Arguments A1 and A3 correspond to the first concern and
arguments A2 and A4 to the second concern. All paninvalid argument patterns
of the kind treated in Prior Analytics fall into one or other of these two classes.
These two concerns are treated in turn immediately below.
Paninvalid patterns relating to a concludent pattern in the standard syntax
Looking back at Aristotle’s treatment of Barbara (PaM, MaS | PaS) in Prior
Analytics A\ we notice that (1) he did not demonstrate its panvalidity but posited
this pattern as obviously panvalid and (2) he did not demonstrate the paninvalidity
of the argument pattern PaM, MaS | PeS, which has the same premiss-pair pattern
as Barbara. Nor, for that matter, did he show that PaM, MaS | P iS is panvalid
and that PaM, MaS | PoS is paninvalid. All these argument patterns have the
same concludent premiss-pair pattern. Since Aristotle does not specially take up
these concerns, we interpolate from the text to illuminate his thinking. We first
consider the premiss-pair pattern represented in argument A1 from which Barbara
results and then consider the premiss-pair pattern represented in argument A3
from which Darii results.
Take the following two arguments in the first figure, the one substituting ‘ani¬
mal’, ‘mammal’, ‘human’ for belonging to every, the other ‘animal’, ‘reptile’, ‘snake’
for belonging to none.
Since “Animal belongs to every man” is recognized to be a necessary result,
Aristotle understood that ‘no other syllogistic result is logically possible’. Thus,
PaM, MaS | is a concludent premiss-pair pattern. Having posited that an a sen-
204
George Boger
Table 31.
Pattern: PaM, MaS|PaS
1. Animal [A] belongs to every mammal [M].
AaM
T
2. Mammal belongs to every human [H].
MaH
T
? Animal belongs to every human.
AaH
F
Pattern: PaM, MaS|PeS
1. Animal [A] belongs to every reptile [Rj.
AaR
T
2. Reptile belongs to every snake [S].
RaS
T
? Animal belongs to no snake.
AeS
F
tence is a necessary result of sentences in the given pattern PaM, MaS | (A4 :
25b37-40), Aristotle would have immediately recognized that an e sentence could
never follow logically: this satisfies Condition I — granting the one sentence ex¬
cludes the other. It is logically impossible to find terms for PaM, MaS | PeS
where both the premisses and the conclusion are true sentences. He might have
added that it is logically impossible to derive something not belonging universally
from something belonging universally. In the case of Celarent (PeM, MaS [ PeS)
it would be impossible to have a true a sentence as a conclusion when its corre¬
sponding e sentence is true. 83 Recognizing that for Aristotle it is trivially true that
a partial sentence follows logically from a universal sentence of the same quality,
we see that he would have determined that PaM, MaS | PiS is panvalid and that,
likewise, PaM, MaS | PoS is paninvalid. Aristotle recognized that what happens
here is similar to what happens in applying the method of deducing inconcludence
from the indeterminate. The reasoning is the same for any syllogistic pattern in
each figure whose conclusion pattern is universal.
Next consider the case when the result in the first figure is partial. The following
two arguments illustrate that the premiss-pair pattern PaM, MiS | is concludent;
substitute ‘surface’, ‘table’, ‘furniture’ for belonging to every , ‘container’, ‘bottle’,
‘plastic’ for belonging to none.
It is logically impossible not to satisfy Condition I for this premiss-pair pattern
even though the necessary result is a partial sentence. Again, take the second
figure pair pattern MeP, MeS | from which Aristotle gets Festino, MeP, MiS |
PoS, and substitute terms for belonging to every and belonging to none as below.
It is logically impossible to satisfy Condition II for this pattern.
However, in a concludent pattern where the necessary conclusion is partial, con¬
trary to a concludent pattern whose necessary result is universal, it is possible to
exhibit two arguments each with true premisses and a false conclusion. For MeP,
MiS | PaS take the instance cited immediately above and for MeP, MiS | PeS
take the following instance (Table 34):
83 Applying the method of fact in this instance, a method surely known to Aristotle (see B2 ),
we can demonstrate for ourselves that PaM, MaS | PeS is a paninvalid argument pattern, as
also is the pattern with a weakened conclusion (PaM, MaS | PoS).
Aristotle’s Underlying Logic
205
Table 32.
Pattern: PaM, MiS|PaS
1. Surface [S] belongs to every table [T].
SaT
T
2. Table belongs to some furniture [F].
TiF
T
? Surface belongs to every furniture.
SaF
F
Pattern: PaM, MiS|PeS
1. Container [C] belongs to every bottle [B]
CaB
T
2. Container belongs to some plastic [P].
BiP
T
? Container belongs to no plastic.
CeP
F
Table 33.
Pattern: MeP, MiS|PaS
1. Bi-pedal [B] belongs to no turtle [T].
BeT
T
2. Bi-pedal beongs to some animal [A].
BiA
T
? Turtle belongs to every animal.
TaA
F
Pattern: MeP, MiS|PeS
1. Intelligence [I] belongs to no building [B]
IeB
T
2. Intelligence belongs to some animal [A].
HA
T
? Building belongs to no animal.
CeP
F
This signals that both a universal privative and a universal affirmative sentence
cannot result from the concludent pattern MeP, MiS |. Here again we recognize
Aristotle’s familiarity with the method of fact (Pr. An. B2-4). Moreover, it is
logically impossible to generate a true attributive sentence from a true privative
sentence and a true universal from a true partial sentence. Having eliminated a and
e sentences as possible results, Aristotle would have turned to i and o sentences to
determine which is necessary. We know that Aristotle established the panvalidity
of Festino by a metalogical deduction using Ferio (27a32-36). Recognizing that
nothing affirmative can result from something privative, he would eliminate an
i result. This leaves MeP, MiS | PoS as the only possible logically necessary
result. 84
Paninvalid patterns from a concludent pattern with conclusion conversion
This concern relates to altering the standard syllogistic syntax by converting the
conclusion pattern — from PrrS to SiP — while retaining the syntax of the
premiss-pair pattern. Thus we have:
84 Perhaps Aristotle reasoned in this way. This seems likely from his using the method of
deducing inconcludence from the indeterminate. Perhaps, he deduced the panvalidity of MeP,
MiS | PoS from the panvalidity of MeP, MaS | PeS (Cesare), a strong pattern, which was itself
likely established by fiddling with conversions and premiss transposition of Celarent (PeM, MaS
| PeS).
206
George Boger
Table 34.
Pattern: MeP, MiSjPaS
1. Bi-pedal [B] belongs to no turtle [T].
BeT
T
2. Bi-pedal beongs to some animal [A].
BiA
T
? Turtle belongs to every animal.
TaA
F
Pattern: MeP, MiS|PeS
1. Good [G] belongs to no danger [D]
GeD
T
2. Good belongs to some house [H],
GiE
T
? Danger belongs to no house.
DeH
F
1. PzM
1. MzP
1. PzM
2. Mt/S
2. MyS
2. SyM
? SzP
? SzP
? SzP
This appears to double the possible results of premiss-pair patterns. If Aristotle
specifically treated premiss-pair patterns and not arbitrary arguments — and thus
did not use the method of counterargument — how would he have established the
paninvalidity of an argument pattern whose premiss pattern has been determined
to be concludent but whose conclusion is the converse of the standard syntax?
Moreover, do any panvalid argument patterns emerge from inconcludent premiss-
pair patterns when the conclusion is converted?
First, we may easily dispense with all concludent patterns whose necessary re¬
sults are either e or i sentences because, involving simple conversion, panvalidity
is preserved. This, of course, also applies to inconcludent patterns with e or i con¬
clusions; paninvalidity in such cases is preserved. But what happens in the cases
of an a or an o conclusion, say, in an argument fitting the premiss pattern of Bar¬
bara but with a converted conclusion, PaM, MaS | SaP, or one fitting the premiss
pattern of Bocardo, PoM, SaM | SoP? In each of these two cases, representative
of a and o sentences, panvalidity is ruled out. Neither an a sentence nor an o
sentence admits of simple conversion: the a converts per accidens , the o does not
convert. Thus, if either an a or an o sentence is a necessary result of a given con¬
cludent premiss-pair pattern, then, since neither kind of sentence converts simply,
no argument fitting a concludent pattern whose conclusion is a conversion of the
standard syntax is valid. These argument patterns are accordingly determined
necessarily to be paninvalid. We need only refer to Prior Analytics A2 on the
conversion rules to grasp Aristotle’s reasoning. 85
Finally, it is not possible, by altering the standard syllogistic syntax by convert¬
ing the conclusion pattern, to generate a panvalid argument pattern, that is, that
a given inconcludent pattern become concludent. A given premiss-pair pattern is
85 In Pr. An. B22 Aristotle treats the matter of conversions but in a confusing way. R. Smith
(1989: 216-219) is not happy with Aristotle’s treatment of the matter and notes that in some
ways he repeats what he accomplishes at B5- 7.
Aristotle’s Underlying Logic
207
either concludent or inconcludent. Notice that Aristotle encounters this matter
with Camestres in the second and with Disamis in the third figure where in each
instance he converts the derivation to re-establish the standard syntax. This indi¬
cates that for second and third figure argument patterns a conclusion conversion
amounts to transposing the premisses and converting the major to the minor and
the minor to the major term to re-establish the standard syllogistic syntax. In
truth, doing this for second and third figure argument patterns amounts to treat¬
ing every possible arrangement of two categorical sentences taken as premisses in
those two figures. Consider, for example, the relationship between Cesare and
Camestres and that between Disamis and Dimaris.
The same reasoning applies to the first figure in relation to the purported ex¬
istence of a fourth, or indirect first, figure. Examining why this is so might help
to reveal why Aristotle considers there to be only three figures. Each so-called
fourth figure argument pattern with at least one convertible sentence pattern as
a premiss is analyzable ( Pr. An. AJ,5) into a second or third figure pattern: into
the second figure when the convertible sentence pattern is the minor premiss; into
the third figure when the convertible sentence pattern is the major premiss. This
accounts for 12 of the 16 possible premiss-pair patterns in the first figure. The
patterns among these combinations are: Dimaris [Disamis (2)]; Fresison [Festino
(2) or Ferison (3) or even Ferio (1)]; Camenes [Camestres (2) and with a weakened
conclusion Camenop (Camestrop [2])]; and Fesapo [Felapton (3)] (see Parry 1991:
282-287). This leaves MaP, SaM |, MaP, SoM |, MoP, SaM |, MoP, SoM |. We
treated MaP, SaM | above (this section) and saw that this pattern amounts to
a simple conversion of the conclusion of Barbara, which is not logically possible.
However, converting the conclusion of Barbara per accidens , which is logically
possible, and transposing premisses produces Bramantip (similar to Darapti) —
Barbarix is its counterpart in the first figure. Finally, it is clear to Aristotle that
MaP, SoM |, MoP, SaM |, and MoP, SoM | amount to a premiss transposition re¬
quired by converting the conclusions of first figure patterns. The most that might
be obtained, then, is an o sentence. However, since an o sentence is not obtainable
from the counterpart of these premiss-pair patterns in the first figure, these con¬
clusion conversions, with their concomitant premiss transpositions, would also not
be logically possible. Thus, seeing that Aristotle considers premiss-pair patterns,
as discussed above, we can grasp his considering there to be only three figures. 86
4 REFINING THE SET OF SYLLOGISM RULES
4-1 Establishing independence among deduction rules
As we remarked at the outset of this study (§1.2), the three different interpre¬
tations of Aristotle’s logic are in significant agreement about the place of reduc¬
tion in his system. In fact, they tend to consider the processes of completion
86 L. Rose’s discussion (1968: 57-79) of the fourth figure is very instructive.
208
George Boger
( teleiosis, teleiousthai) , reduction (dvaywyf] [anagoge], dvdyetv [ anagein ]), and
analysis (otvaXuau; [ analusis ], avaXuetv [analuein]) to be virtually identical. 87 It
is peculiar that such different interpretations of a syllogism could produce such
similar views about the logical relationships among them. If we allow for con¬
flating an argument pattern or form (one traditionalist sense of ‘syllogism’) and
a corresponding conditional sentence expressing such a pattern (the axiomaticist
position), the similarity becomes more apparent. 88 The various interpreters hold
that reduction amounts to deduction of some syllogisms, taken as derived, from
others, taken as primitive, to form a deductive system. The traditionalist R. M.
Eaton, for example, holds that reduction shows that “the validity of these [sec¬
ond and third figure] moods is deducible from that of moods in the first figure”
(Eaton 1959: 123; author’s emphasis) and that in general reduction is a process
of transforming syllogisms (Eaton 1959: 86, 90, 126). J. Lukasiewicz (1958: 76;
cf. 43-44) expresses an axiomaticist view that “reduction here means proof or
deduction of a theorem from the axioms”; reduction is an indispensable process
of deriving syllogistic theorems from axioms using an implicit propositional logic.
J. Corcoran, a deductionist, writes in a similar vein that “‘reduce to’ here means
‘deduce by means of’”(1974: 114; author’s emphasis). 89
However, when we consider Aristotle’s treatment of reduction at Prior Ana¬
lytics A 7 we discover that he distinguishes the process of completion from that
of reduction: “it is also [xou.] possible to reduce all the syllogisms to [avayayeiv
iravxac; xou<; auXXoyiapou^ eu;] the universal syllogisms in the first figure” (29b 1-2;
cf. 29a30-29b2). Aristotle expressed the distinction more forcefully at A 23.
It is clear from what has been said, then, that the syllogisms in these
figures [viz. the second and third figures] are both completed by means
87 J. Lukasiewicz, for example, explicitly considers ‘anagein’ (to reduce) and ‘analuein’ (to
analyze) to be synonyms (1958: 44), and this is the case also with R. Smith (1989: 161), J.
Corcoran (1981: 6), J. W. Miller (1938: 25), and L. Rose (1968: 55). It has been customary in
all three interpretations to conflate analysis, reduction, and completion. Consider, for example,
W. D. Ross’ commentary on .47 in relation to A4-6 (1949: 314-315) and on A45 (1949: 417-
418); J. N. Keynes’ (1906: 318-325) and R. M. Eaton’s (1959: 86, 90, 122-124, 128-131) remarks
that transforming and deducing syllogisms amount to reduction; and G. Patzig’s (1968: 134-137)
similar position. Both Patzig (1968: 135) and Eaton (1959: 109) refer to the second and third
figure syllogisms as being “disguised” first figure syllogisms. See also Smith’s commentaries on
A 7 (1989: 118-119; cf. 1986: 58-59), on A23 (1989: 141), and on A45 (1989: 177).
88 Consider A. N. Prior’s assessment of J. Lukasiewicz’s thinking (1955: 116-117) and J. W.
Miller’s treatment of the system (1938: ch. 3; cf. 11-14). Miller, in fact, writes (1938: 14,
25, 28) that he is applying the postulational method of Aristotle himself and “merely carrying]
to its completion an undertaking which Aristotle himself began”. Also consider W. T. Parry’s
treatment of Aristotle’s “deductive system” (1991: ch. 20; cf. 520n2, 521n9) and J. Corcoran’s
remarks (1983) on a connection between a Gentzen-sequent natural deduction system and J.
Lukasiewicz’s axiomatic system.
89 J. Corcoran refers to Pr. An. AT: 29bl-2 and cites it as follows (1974: 114): “It is possible
also to reduce all syllogisms to the universal syllogisms in the first figure”. Note Corcoran’s
change of position on the matter of reduction as expressed in 1974 and later in 1981 and 1983.
His earlier position did not especially vary from either the traditionalists’ or T. Smiley’s views.
Later he considered reduction to be “heuristically perverse” insofar as determining validity is
concerned (1981: 4-5).
Aristotle’s Underlying Logic
209
of [xsXetouvxoa te 8ia] the universal syllogisms in the first figure and
reduced to them [xou etc; toutouc; avdyovTou]. (40bl7-19)
The topic of reduction is introduced and concluded at A1 by using the verb
l anagein\ which we translate by ‘to reduce’. What seems to have confused in¬
terpreters is Aristotle’s treating the reduction of syllogisms by using the verbs
‘ teleiousthai ', ‘ epiteleisthai', and ‘hEixvuoOai’ (deiknusthai ), but not ‘ sullogizesthai’,
exactly as he treats the syllogisms at A 5-6. Here is what he writes in A 7, for ex¬
ample, about second figure reduction.
It is evident that those in the second figure are completed [xeXetoOvTai]
by means of these [universal first figure] syllogisms, although not all
in the same way; the universal syllogisms are completed when the
privative premiss is converted, but each of the partial syllogisms is
completed through leading away to an absurdity. (29b2-6; see 29b6-8
for the first figure and 29bl9-24 for the third figure)
Indeed, Aristotle provides only two actual illustrations of reduction, those of Darii
(29b8-ll) and Ferio (29b 11-15), and these reductions are expressed in exactly the
same manner as the deductions in A5-6 , even using the language of completion.
Thus, some logicians have taken him not to distinguish two processes but to du¬
plicate in A7 the project of A 4 - 6 . However, Aristotle is not here concerned to
demonstrate the truth or falsity of a given sentence nor the validity or invalidity of
a given argument. Nor is he concerned here to show that a given argument pattern
is panvalid as at A5-6. Rather, he is now concerned to demonstrate that any sen¬
tence fitting any one of the four categorical sentence patterns (i.e., any problema)
can be established to be a logical consequence (conclusion) of other categorical
sentences by using only the two patterns of the universal teleioi sullogismoi as
deduction rules. To do this, it is true, he performs deductions as he does at A5-6,
or he refers to those already performed there. But now he has a different objective
in examining the relationships among the patterns which are used by him as rules:
namely, to simplify his deduction system. 90
Aristotle first treats the reduction of the four second figure panvalid patterns,
and he treats them in a manner that suggests his readers’ familiarity (29b2-6).
Again, he does not perform deductions here but refers us to A5 where he had
already established their panvalidity by using the first figure teleioi sullogismoi
90 Consider, for example: On Generation and Corruption 330a24-25 on reduction of contraries
to two pairs, hot-cold, dry-moist (cf. Physics 189b26-27, Meta. 1004b27-1005a5, 1036b21-
22, 1061al-2, 13-14); Movement of Animals 700bl8-19 on reducing to thought and desire such
sources of movement as intellect, imagination, appetite, etc. We can illuminate Aristotle’s use of
‘anagein’’ at A 7 by citing a rather exact analogue in his discussion of locomotion in Physics 7.2.
There Aristotle writes that all forms of locomotion caused by something other than the object
in motion are reducible to (dvayovTai et?) four kinds, namely, to pulling, pushing, carrying, and
twirling (243al6-18); and he even reduces (dvayrtv) carrying and twirling to pulling and pushing
(243bl5-17). There are numerous instances of similar usage in Aristotle’s writings, all of which
concern identifying what amount to the special principles (ISiai dpyod) of a given subject matter.
210
George Boger
patterns, Barbara and Celarent, as rules in the completion process. He writes
only that:
It is evident that those in the second figure are completed through these
syllogisms, although not all in the same way; the universal syllogisms
are completed when the privative premiss is converted, but each of the
partial syllogisms is completed through leading away to an absurdity.
(29b2-6)
At A7 both Festino and Baroco are completed using reductio proof, where in A5
only Baroco was treated this way. With this established he could proceed to the
two partial patterns of the first figure, Darii and Ferio, which are reduced through
second figure patterns. Since, in order to show that only the two universal first
figure patterns are sufficient, he spends considerably more time on the reduction
of the two partial patterns (29b6-19) of the first figure. He concludes (29bl9-
25) with the reduction of the six third figure patterns in the same manner as he
treated the second figure patterns. Table 36 summarizes what Aristotle writes
about reduction. Table 35 provides their deductions (completions).
Aristotle’s Underlying Logic
211
Table 35. Summary of Aristotle’s texts on each panvalid pattern in his treatment
of reduction
Pattern con¬
sidered
Manner of the completion
Pattern
used in the
completion
Barbara:
[completed ( teleiousthai ) probatively
through itself 5t oturou].
Barbara
Celarent:
[completed probatively through itself].
Celarent
Camestres:
completed probatively dia Celarent (29b2-
6; cf. A5, 27a9-14).
Celarent
Cesare:
completed probatively dia Celarent (19b2-
6; cf. A5, 27a5-9).
Celarent
Festino:
completed by leading to an impossibility
( reductio ) dia Celarent (19b2-6; cf. A5,
27a32-36).
Celarent
Baroco:
completed by reductio dia Barbara (29b8-
11, 15-19)
Barbara
Darii:
proved (deiknusthai ) by reductio dia
Camestres;
Camestres completed probatively dia
Celarent (19bll-15, 15-19).
Celarent
Ferio:
proved by reductio dia Cesare;
Cesare completed probatively dia Celarent
(29bl 1—15, 15-19).
Celarent
Darapti:
Completed probatively dia Darii;
Darii proved by reductio dia Camestres;
Camestres completed probatively dia
Celarent (19b21-25; cf. A6, 28al7-22).
Celarent
Datisi:
same as Darapti (19b21-25; cf. A6, 28bll-
15).
Celarent
Disamis:
same as Darapti (19b21—25; cf. A6 , 28bll-
15).
Celarent
Felapton:
completely probatively dia Ferio;
Ferio proved by reductio dia Cesare; Ce¬
sare completed probatively dia Celarent
(19b21—25); cf. A6 : 28a26-30).
Celarent
Ferison:
same as Felapton (29b21—25; cf. A6,
28b33-35).
Celarent
Bocardo:
completed by reductio dia Barbara (29b21-
25; cf. A6, 28b 17-20).
Barbara
Table 36. Completions relating to Aristotle’s reduced deduction system
Barbara
Celarent
Darii 29b6-19
Ferio 29b6-19
1. AaB
1. AeB
1. AaB
1. AeB
2. BaC
2. BaC
2. BiC
2. BiC
? A aC
? AeC
? AiC
? AoC
3. AaC
1,2 Barb 3. AeC
1,2 Celar 3. AeC
assume
3. AaC
assume
4. CeA
3e-con
4. BeA
1 e-con
5. AaB
1 rep
5. AaC
3 rep
6. CeB
4,5 Celar
6. BeC
4,5 Celar
7. BeC
6 e-con
7. BeC k BiC
6,2 conj; X
8. BeC k BiC
7,2 conj; X
8. AoC
3-7 reduct
9. AiC
3-8 reduct
Camestres
Cesare
Festino
Baroco
1. BaA
1. BeA
1. BeA
1. BaA
2. BeC
2. BaC
2. BiC
2. BoC
? AeC
?AeC
? AoC
?AoC
3. CeB
2 e-con
3. AeB
1 e-con
3. AaC
assume
3. AaC
assume
4. BaA
1 rep
4. BaC
2 rep
4. BeA
1 rep
4. BaA
1 rep
5. CeA
3,4 Celar
5. AeC
3,4Celar
5. AaC
3 rep
5. AaC
3 rep
6. AeC
5 e-con
6. BeC
4,5 Celar
6. BaC
4,5 Barb
7. BeC k BiC
6,2 conj; X
BaC& BoC
6,2 conj; X
3-7 reduct
8. AoC
3-7 reduct
8. AoC
212 George Boger
Darapti
1. AaB
2. CaB
? AiC
3. AeC
4. CeA
5. AaB
6. CeB
7. CeB&
8. AiC
Datisi
Disamis
Felapton
1. AaB
1. AiB
1. AeB
2. CiB
2. CaB
2. CaB
? AiC
? AiC
? AoC
assume
3.
AeC
assume
3. AeC
assume
3. AaC
assume
3 e-con
4.
CeA
ee-con
4. CaB
2 rep
4. BeA
1 e-con
1 rep
5.
AaB
1 rep
5. AeB
3,4 Celar
5. AaC
3 rep
4,5 Celar
6.
CeB
4,5 Celar
6. AeB & AiB
5,1 conj; X
6. BeC
4,5 Celar
6,2 conj; X
7.
CeB & CiB
6,2 conj; X
7. AiC
3-6 reduct
7. CeB
6 e-con
3-7 reduct
8.
AiC
3-7 reduct
8. CeB k CaB
7.2 conj; X
9. AoC
3-8 reduct
Ferison
Bocardo
1. AeB
1. AoB
2. CiB
2. CaB
? AoC
? AoC
3. AaC
assume
3. AaC
assume
4. BeA
1 e-con
4. CaB
2 rep
5. AaC
3 rep
5. AaB
3,4 Barb
6. BeC
4,5 Celar
6. AaB k AoB
5,1 conj; X
8. CeB & CiB
7,2 conj; X
9. AoC
3-8 reduct
>
’-I
w’
O
5T
Cfl'
3
a
3
Gq
r
O
CFP
to
CO
214
George Boger
By treating each case of a panvalid pattern — and this amounts to treating all
possible combinations of concludent premiss-pair patterns — Aristotle established
the deductive preeminence of the patterns of the two universal syllogisms, Barbara
and Celarent, as the only syllogism rules necessary in his deduction system. He
demonstrated not only that second and third figure patterns are redundant de¬
duction rules, which was implicitly established at Prior Analytics A4-6 , but also
that the two partial patterns of the first figure are equally redundant. The same
deductive results are accomplished using only the universal patterns of the first
figure. Thus, Aristotle’s reduction of syllogistic patterns at A 7 is not a substitute
for syllogistic deduction nor a process for axiomatizing a system of logic as ax-
iomaticists hold, but a metalogical process for establishing the independence of a
small set of deduction rules.
4-2 Analysis distinct from reduction
Aristotle treated analysis at Prior Analytics A45. At the outset of A45 he used
‘anagein’ as a synonym for 'analuein ’ (50b5-9) in a way apparently inconsistent
with what he wrote at A 7 and A 23: “not all syllogisms can be reduced ... but only
some [oux aroxvxac; 5s ak\’ svtoix;]”. Recall that at A 7 he showed that all the syllo¬
gisms could be reduced. We are thus struck by an apparent limitation announced
here. However, while Aristotle treated the analyses of first into second and second
into first figure syllogisms (50b5-30) by using 1 anagein’, he abruptly switched at
50b30-32 (re Baroco) to use ‘ analuein ’ and continued to do so throughout A45
respecting all the other possible analyses (50b30-51a39). In addition, Aristotle
clearly distinguished analysis from reduction at A45 as he distinguished reduction
from completion at A 7: “... it is evident how one must reduce syllogisms [tcCk;
pev ouv Sei root; auXAoyiapou<; avaysiv], and that the figures are analyzed into one
another [xal oti avaXuexca xa oy^paxa ei<; aXXqXa]” (51b3-4).
Aristotle established the scope of analysis in the following way. While every syl¬
logism is reducible (to) (29bl-2), not every syllogism is analyzable (into). Passages
in A45 where Aristotle specifically stated that no analysis of one syllogism into
another is possible include: 50b8-9, 50bl8, 50b30-32, 50b33-34, 51al-3, 51al8-19,
51a27, 51a31-32, 51a37-39, and 51a40-41. This is sufficient to establish a difference
between the two processes. By examining each case, we can extract Aristotle’s two
rules for analyzing syllogisms.
1. A given syllogism in one figure is analyzed into a syllogism in another figure
whenever both syllogisms prove the same problema, that is, each syllogism
proves a sentence fitting the same pattern, whether an a, e, i, or 0 sentence
pattern (50b5-8).
Aristotle is writing metalogically here. Thus, for example, there is no analysis of
a syllogism fitting Barbara into a syllogism fitting a pattern in any other figure,
nor is there of one fitting Darii into one fitting a second figure pattern.
Aristotle’s Underlying Logic
215
2. A given syllogism fitting a pattern in one figure is analyzed into a syllogism
fitting a pattern in another figure by using only conversion and premiss
transposition (51a22-25).
Thus, for example, neither a syllogism fitting Baroco nor one fitting Bocardo can
be analyzed.
Aristotle treated the possible analysis of almost every syllogism and he identified
which are not analyzable, either because conversion does not produce a syllogism
or because the same problema is not proved in each figure. Table 37 summarizes
Aristotle’s results at Prior Analytics A45. 91
Table 37.
Summary of the analyses treated in Prior Analtyics A45
Barbara
No analysis possible
Celarent
Into Cesare (50b9-13)
nto Camestres (not treated) 0
Darii
Into Disamis (not treated) 0
Into Datisi (50b33-38
Ferio
Into Festino (50bl3-16)
Into Ferison (50b38-40)
Cesare
Into Celarent (50bl7-21)
Camestres
Into Celarent (50b21-25) c
Festino
Into Ferio (50b25-30)
Into Ferison (51a26-30)
Baroco
No analysis possible
Darapti
Into Darii (51a3-7
Felapton
Into Ferio (51al2-15)
Into Festino (51a34-37)
Disamis
Into Darii (51a8-12) c
hline Datisi
Into Darii (51a7-8)
Boardo
No analysis possible
Ferison
Into Ferio (51al5-18)
Into Festino (51a34-37)
Aristotle treated analysis in Prior Analytics A45 quite differently from reduc¬
tion in A1 and from completion at A 5-6, both of which involve a deduction process.
Here there is no direct concern with a process of deduction to show that a given
premiss-pair generates a syllogism nor that a conclusion follows from premisses
91 Table 37 uses the following notation: superscript c — premiss transposition and conclusion
conversion.
Aristotle treats neither analyzing Darii and Disamis into each other nor Celarent and Camestres
into each other. This is simply an oversight.
216
George Boger
nor that some patterns are redundant as deduction rules. Neither probative nor
reductio proofs are cited in relation to analysis, although Aristotle always pre¬
served strict syllogistic syntax through conversion in the case of some conclusion
patterns.
Characteristically Aristotle conceived of analysis as one syllogism being trans¬
formed into another. In fact, he referred to the process of analysis as a fiETOtpaaic;,
a transition or transformation (51a24-25). Analyses of syllogisms occur between
any of the figures. Thus, no syllogistic pattern nor any figure has preeminence in
relation to analysis as is the case with completion and reduction. That Aristotle
used ‘ anagein ’ in an apparently contrary way at the outset of AJ t 5 is mitigated
when once we view the texts of A7 and AJ,5 more globally. Aristotle unequivocally
distinguished the process of reduction from that of analysis. 92
5 CONCEPTS IN ARISTOTLE’S LOGIC
While modern logicians believe that Aristotle developed a logic that contains a
notion of formal deducibility — and their mathematical models establish that this
is so — they do not believe that he explicitly formulated this notion, either in
general of deductive systems or specifically of his syllogistic system. In addi¬
tion, it might seem that Aristotle did not define ‘following from necessity’, to s£
dvayxr^ auppatvetv, his expression for logical consequence, and then show that the
syllogisms are true to it. Previous interpreters believed that he only posited the
four teleioi sullogismoi and then ‘reduced’ the validity of the others to them as
‘principles’. However, modern interpreters have tended not to place Aristotle’s
logical investigations in Prior Analytics into the larger context of his other works
relating to logic. Surely what he wrote in Prior Analytics is the product of con¬
siderable intellectual exploration. Accordingly, then, we might expect to find his
thinking about logical consequence elsewhere in the larger corpus and then piece
together an account that accommodates modern logicians. In particular, what
Aristotle expressed about logical necessity in Metaphysics and in Prior Analytics
comports exactly with his treatment of deducibility in Prior Analytics. Below we
first extract Aristotle’s syntactic notion of formal deducibility (§5.1) and then his
semantic notion of logical consequence (§5.3). Two kinds of deduction also are
represented (§5.2).
5.1 Aristotle’s notion of formal deducibility
The modern notion of formal deducibility, or formal derivability, can be stated as
follows:
92 While it may seem unbecoming to take Aristotle’s expression at 50b5-30 as a lapse in precision
not later amended, we believe this to be so and that our interpretation does not do violence to
Aristotle’s meaning.
Aristotle’s Underlying Logic
217
A given sentence c is formally deducible from a given set of sentences
P when there exists a finite sequence of sentences that ends with c and
begins with P such that each sentence in the sequence from P is either
a member of P or a sentence generated from earlier sentences solely by
means of stipulated deduction rules.
The salient features of this notion pertain to a chain of reasoning from premisses to
conclusion such that the chain sequence (1) is finite and (2) might use (a) repetition
and (b) stipulated deduction rules. In addition such a chain is cogent in context.
Now, it is evident from his discussions in Prior Analytics and Posterior Analytics
that Aristotle subscribed to a notion of formal deducibility remarkably similar to
this modern formulation. Moreover, while Aristotle did not express this notion
in one rigorously constructed sentence, he nevertheless provided many statements
in Prior Analytics and in Posterior Analytics from which we can extract his own
understanding. Here again we take his several statements and organize them
according to a modern practice.
Some logicians believe that the closest Aristotle came to defining ‘deduction’
per se is his definition of ‘syllogism’ at the outset of Prior Analytics. He writes
that a syllogism ( sullogismos ) is
a discourse [Adyop] in which, certain things having been supposed
[posited or taken], something different from the things supposed [posited
or taken] follows of necessity because these things are so [exepov xt xwv
XEtfiEvcov sc; dvayxr]c; aup(3dtvEt xw xauxa livou]. (24bl8-20) 93
Aristotle immediately continues by defining certain aspects of this definition.
By ‘because these things are so’ I mean ‘resulting through them [xo Sid
xauxa aujapdiveiv]’, and by ‘resulting through them’ I mean ‘needing
no further term from outside in order for the necessity to come about
[xo jjqSevoc; e^coOev opou xpoahciv ttpot; xo ysveaBat xo avayxaiov]’.
(24b20-22)
However, Aristotle seems here rather more to define ‘valid argument’ than ‘deduc¬
tion’; there is no strong indication of an epistemic process present in a syllogism.
We have already seen that he understood his project in Prior Analytics A5-6
as establishing which patterns of two categorical sentences when taken together
as premisses result in a syllogism. Aristotle’s logical methodology is to perform
deductions in the metalanguage of Prior Analytics to establish that certain argu¬
ment patterns are panvalid patterns. 94 He then proceeds to use these patterns in
a deduction process. This being so, we conclude that a syllogism, at least in this
93 Aristotle defines ‘syllogism’ in much the same way at three other places: Top. 100a‘25-27,
SR 164b27-165a2, and Rh. 1356bl6-18.
94 In this connection, consider the following passages: in Pr. An. A5: 27al2-13, 14, 14-15,
16-18, 36, 27b3, 28a2-7; in A6: 28a22, 22-23, 28-30, 28bl3-15, 20-21, 34-35, 29al5-16.
218
George Boger
discussion, is a valid argument, one that might, to be sure, be used in a deduction
process, but which is not itself a deduction. Again, we understand him as securing
a set of deduction rules. The corroborating evidence for this interpretation is Aris¬
totle’s distinguishing syllogisms into those that are teleios, usually translated by
‘complete’ or ‘perfect’, and those that are ateles , usually translated by ‘incomplete’
or ‘imperfect’; ‘ dunatos ’, translated by ‘potential’, serves as a synonym for ‘ateles’.
His definitions of ‘ teleios ’ and ‘ateles' in Prior Analytics A1 and his references to
the teleioi sullogismoi help to secure our interpretation.
I call a syllogism complete [teXetov] if it stands in need of nothing else
besides the things taken in order for the necessity to be evident [xpoc;
to cpavf)vou to dvayxa lov] ; I call it incomplete [otTeXf}] if it still needs
either one or several additional things which are necessary because of
the terms assumed, but yet were not taken as premisses. (24b22-26)
Of course, ‘ teleios ’ and ‘ ateles ’ are epistemic notions and refer to evidency of va¬
lidity. From Prior Analytics AJ,, A5-6 , and A 7, then, we notice that Aristotle,
besides using two formal processes of deduction (direct and indirect; §5.2), also
identifies two degrees of someone’s recognizing validity, that is, in connection with
the syllogisms, of experiencing a mental act by which someone grasps that ex¬
treme terms are mediated by a middle term. (1) A syllogism being ateles means
that a participant has to go through a number of deductive steps, more than one,
to recognize its validity. Being dunatos means that it is possible to recognize its
validity. (2) A syllogism being teleios means that a participant has to go through
only one step to recognize the validity. At places Aristotle helps us to understand
the meaning of ‘ teleios ’ by referring to the evidency of the necessity of the conclu¬
sion following logically from the premisses of a first figure syllogism as ‘Si auTou’,
or through itself. In Prior Analytics A 7 he writes of the two partial syllogisms
of the first figure (viz., Darii and Ferio), and we interpolate to include the two
universal syllogisms (Barbara and Celarent) as well: “ the partial syllogisms in
the first figure are brought to completion through themselves [ol 5’ ev tw xpwTco, oi
Korea pepot;, eTtiTeXouvTai pev xai. 51 aOrcov]” (29b6-7) 95
Recognizing validity in the case of a first figure syllogism requires only one step,
more steps in the case of a second or third figure syllogism, and, of course, even
more steps in a case of a valid argument with more than two premisses. Thus,
the manner of Aristotle’s discussion in Prior Analytics A5-6 and A7 shows that
he understands the process of deduction to establish, or to make evident in the
mind of a participant, that a given argument is valid — or, as the case may be,
that a given argument pattern is panvalid. And, moreover, he uses the syllogism
95 In contrast, consider: “all [second figure syllogisms] are completed [eiuteXoGvcci!.] by taking
in addition certain things”(.Pr. An. A5: 28a5-6); “all [third figure syllogisms] are completed
[teXeiouvtcxi] by taking certain things in addition”(,4<i: 29al5-16); and “all [first figure syllogisms]
are completed by means of the things initially taken [jt&vtst; yap etuteXouvtou 8ia twv e? apxr)<;
Xr]<p0EVT(ov]”(Pr. An. A4‘- 26b28-33)] or “are completed through themselves” ( Pr. An. A7-.
29b6-8).
Aristotle’s Underlying Logic
219
patterns in this process. We can illuminate this with the following illustrations
(Table 38). We refer to two-premiss arguments, but this applies mutatis mutandis
to one-premiss arguments.
Table 38.
1
2
3
4
1. Animal [A] be-
1.
AaM
1.
PaM
1.
AaM
longs to every
mammal [M].
2. Mammal be-
2.
MaD
2.
MaS
2.
MaD
longs to every
dog [D],
? Animal belongs
?
AaD
PaS
?
AaD
to every dog.
3.
AaD 1, 2
Barbara
5
6
7
8
1. Animal [A] be-
1.
AaM
1.
MaP
1.
AaM
longs to every
mammal [M]
2. Animal belongs
2.
AeL
2.
MeS
2
AeL
to no line [L].
? Mammal be-
?
MeL
PeS
?
MeL
longs to no
line.
3.
LeA
2e-conversion
4.
A a
M 1, repetition
5.
LeM
3, 4 Celarent
(Camestres)
6.
MeL
5e-conversion
Item 1 is an object language argument (premisses numbered and the conclusion
indicated by '?’) whose validity, while perhaps obvious, let us take as unknown.
Item 2 is the same object language argument but whose non-logical constants
(terms) have been abbreviated with letters to facilitate recognizing the pattern
of and working with the argument. Item 3 is a metalogical object, in this case
a schematic representation of the panvalid argument pattern named ‘Barbara’. 96
96 Here the letters are schematic letters and the line for the ‘conclusion’ is indicated by the
which signals that a sentence fitting a conclusion pattern has been established always to follow
logically from sentences fitting the premiss pattern. Any semantically precise instance of this
220
George Boger
Item 4 is an object language deduction, albeit quite simple. In this case, the
conclusion of the original argument (item 1) has been established to follow logically,
or to be valid; this is indicated by the fourth line (numbered ‘3’) with the reasoning,
or explanation, provided to the right. Using Aristotle’s nomenclature, item 1 is a
teleios sullogismos. Likewise in this respect, item 5 is an ateles sullogismos and,
thus, accordingly, it requires a deduction to establish its validity (item 8).
Of course, a conversion is not a syllogism according to Aristotle because it
consists in only one premiss. Still, he understood a conversion necessarily to result
in something different. In Prior Analytics Bl, for example, he writes that each
syllogism, save for that whose conclusion is an o sentence, has several different
results:
If A has been proved to belong to every B or to some, then it is also
necessary for B to belong to some A; and if A has been proved to belong
to no B, then neither does B belong to any A (and this conclusion is
different [fe'xepov] from the previous one). (53al0-12)
This, of course, applies also to the subalterns. His taking the conclusions of con¬
versions as different in Prior Analytics A is obvious, if not stated as directly as it
is here. In any case, it is evident that Aristotle took a conversion and a syllogism
equally as species of the same genus, namely, as instances of valid arguments. His
purpose in Prior Analytics is to identify the panvalid argument patterns relating
to converting two terms in a single sentence and to those relating three terms
in two sentences because he recognized the epistemic efficacy of such elementary
patterns in the deduction process — namely, he recognized the rule nature of such
patterns.
We now have some notion of Aristotle’s understanding of deduction, but we
seem to lack his word for ‘deduction’. Jonathan Barnes (1981: 21-25) has sug¬
gested ‘avayxaiov’ ( anankaion ). While Aristotle frequently uses ‘ anankaion ’ as
an adverb, translated by ‘necessarily’, he also often uses it as a substantive, ‘to
avayxaiov’, as in the following passage in Prior Analytics A32 , rendered according
to Barnes.
... that the syllogism is also a deduction [on xod 6 auXXoyiapo<;
avayxaiov screw] since deduction is more extensive than syllogism [ext
xXsov Sc to avayxaiov r] 6 auXXoyiapoi;]; for every syllogism is a de¬
duction, but not every deduction is a syllogism [6 pev yap ouXXoyiapoc;
xap avayxaiov, to S’ avayxaiov ou xav auXXoyiapoc;]. (47a32-35; see
47a31-40)
His use of l to anankaion ’ here seems to indicate that a conversion and a syllogism
are both valid arguments, and perhaps that syllogistic reasoning is only one among
other kinds of deductive reasoning.
argument pattern is a valid argument. This universality, along with its elementary nature, is
why such patterns are employed as rules in a deduction process. Cf. above §3.2.
Aristotle’s Underlying Logic
221
Grasping Aristotle’s understanding of deduction can be enhanced by examining
instances of his performing deductions and by studying the verbs he used to charac¬
terize the deduction process, in particular, the verbs ‘ epiteleisthaV or l teleiousthaf,
as well as ‘KEpdivEaGca’ (perainesthai) , and even ‘ deiknusthai’ . Examining parts
of Prior Analytics A1 helps in this connection. Consider the following passage
(underscoring for comparison).
It is furthermore evident that all [1] the incomplete deductions [2]
are completed through the first figure. For they all [3]
come to a conclusion either probatively or through an impossibility,
and in both ways the first figure [4] results . For those [5] completed pro¬
batively, this results because they all [6] come to a conclusion through
conversion, and conversion [7] produces the first figure. And for those
[8] proved through an impossibility, it results because, when a false¬
hood is supposed, the [9] deduction [10] comes about through the first
figure. (29a30-36)
This is R. Smith’s (1989) translation, which has considerably contributed to rais¬
ing respect for Aristotle’s acumen as a logician. Here is the complete Greek of
Aristotle’s text.
diavEpov 8e xai oxi TtavxE<; [1] ol axeXeu; auXXoytopoi [2] xsXetouvxcu
Scot too TtpwTou ayfijiaToc;. r\ yap SeixtixCk; t) 5lc< too d8uvdxou [3]
TCepouvovTOd. xdvxec dpcpoxepox; 8e [4] ylvExai xo Ttpwxov cryfipa, 5eix-
xlxwc; pev [5] xckeioupevcov, oxi 5id xf)<; dvxLaxpocpf)<; [6] EKEpdivovxo
K&vxe?, f] 8’ dvxiaxpocpf] xo itpoixov [7] etoiei oxrjpa, 5ia 8e xou dSuvaxou
[8] Seixvujjevcov, oxi xeGevxoc xou ([ieuSouc; [9] 6 auXXoyiopo<; [10] ylvExou
8ia xou xpcoxou axf)paxo^.
Now, in light of our comments above, we gloss the text to provide a more faithful
translation, even if awkward, to render Aristotle’s meaning.
It is also clear that all [1] the valid syllogistic arguments whose validity
is not apparent [2] are shown to be valid by means of the first figure.
For, either probatively or by means of reductio, they are all [3] drawn to
a conclusion <by means of a deduction process>. In both cases, <a
syllogism in> the first figure [4] arises <in the deduction process to
establish validity>. Of those whose [5] evidency of validity is estab¬
lished probatively, because they all are [6] drawn to a conclusion
<through a deduction process> by means of conversion, and conver¬
sion [7] produces <a syllogism in> the first figure; of those [8] proved
by means of reductio , because [5] <evidency of validity is established>
by assuming a falsehood [9] a syllogism in the first figure [10] arises
<in the deduction process >.
222
George Boger
In Prior Analytics A7: 29bl-8, as well as in A5-6, Aristotle used ‘ teleiousthai ’
and ‘ epiteleisthai' in exactly the same way: namely, to indicate making evident the
validity of a valid argument specifically by means of generating a teleios sullogismos
in a chain of reasoning. This signals in the mind of a participant, equally today
as then, that the chain of reasoning in which a teleios sullogismos arises is cogent
in context, and thus it links the conclusion sentence as following logically, or
necessarily, from the premiss sentences. It is obvious that Aristotle used the first
figure panvalid patterns as rules in the deduction process to establish knowledge
of validity. While a syllogism is a valid elementary argument, its panvalid pattern
can serve as a deduction rule since every instance is valid. The same thinking
applies to the conversion rules established in Prior Analytics A2 : a conversion is
a valid one-premiss argument, its panvalid pattern serves as a deduction rule.
The elements of syllogistic deductive reasoning, then, consist in three one-
premiss conversion rules and 14 two-premiss syllogism rules, reduced to two in
Prior Analytics A7. Thus, in respect of his syllogistic deduction system, a de¬
duction preeminently involves a chain of reasoning in the mind of a participant
that establishes a conclusion sentence to be a logical consequence of a set of pre¬
miss sentences — a deduction makes validity, or following necessarily, evident. In
the deduction process a participant might use any of the conversion and syllogism
rules as well as repetition, an implicit and often used rule. 97 This is what Aristotle
means in Prior Analytics A25: 42a35-36 by ‘6 Xoyo<; cruXXoyujpot;’, a syllogistic
argumentation, or reasoning syllogistically (ouXXoyioTixwc). In fact, Aristotle is
rather emphatic about this. While it is true that every syllogism has only three
terms and two protaseis, it is just as true that a syllogistic argumentation is not
restricted to two premisses. Rather a syllogistic argumentation consists in chaining
syllogisms that are instances of the rules Aristotle articulated in Prior Analytics
A 4 - 6 . Aristotle used ‘ouoroixia’ ( sustoichia ) to capture the notion of ‘chaining’
syllogisms. 98 And so, just at the places where he restricts a syllogism to two
premisses and three terms, he also writes:
... unless the same conclusion comes about [to au to aupmepaapa ytyviytod.]
through different groups of premisses. (Pr. An. A25: 41b37-38)
... unless something should be taken in addition for the purpose of
completing the deductions [xp6<; tt)v reXeiwaiv twv auXXoyiapcov]. (Pr.
An. A25: 42a33-35)
Aristotle explicitly took up chaining syllogisms in Prior Analytics A25: 42bl-
26, and he provided many examples of this process in both Prior Analytics and
Posterior Analytics. At Prior Analytics A25 he wrote about “counting syllo¬
gisms” and “prior syllogisms” (xpoauXXoyiapoi), about counting terms (opoi),
97 See, e.g., in Pr. An. A5: 27a7-8, 11 and in A6: 28a20-21.
98 LSJ cites ‘series’ and ‘column’ as definitions of ‘cruaxoixlat’. J- Barnes (1994) translates
‘ouaTOixiod by ‘chain’: e.g., Po.An. 79b7, 8-9, 10, 11, 80b27, and 81a21. Also consider Aristotle’s
using ‘ouvajiToc;’, which LSJ defines as ‘joined together’ or ‘linked together’. R. Smith (1989)
translates this by ‘connected’ and ‘dauvourcot;’ by ‘unconnected’: e.g., Pr. An. 41al, 19, 42a21,
65bl4, 33, and 66b27.
Aristotle’s Underlying Logic
223
counting premisses and intervals (itpoTaaeic; and 8L0icmr)[icn:c(), and counting con¬
clusions (oufiTiepdopaxa). We cite Posterior Analytics A25, where Aristotle treats
the superiority of a demonstration having fewer rather than more terms, as an
example of his providing an instance of chaining syllogisms.
... then let one demonstration show that A holds of E through the
middle terms B, C, D, and let the other show that A holds of E through
F, G. Thus that A holds of D and that A holds of E are on a level. But
that A holds of D is prior to and more familiar than that A holds of
E; for the latter is demonstrated through the former, and that through
which something is demonstrated is more convincing than it. (86a39-
86b4; cf., e.g., in Pr. An. A25 : 41b36-42a5 & A28: 44all-44b20)
Thus, we understand that central to syllogistic deductive reasoning is generating
syllogisms as part of an epistemic process. Aristotle summarizes this notion in
Prior Analytics A29.
It is evident from what has been said, then, not only that it is possible
for all syllogisms to come about through this route, but also that this is
impossible through any other. For every syllogism has been proved to
come about through some one of the [three] figures stated previously,
and these cannot be constructed except through the things each term
follows or is followed by (for the premisses and the selection of a middle
is from these, so that it is not even possible for a syllogism to come
about through other things). 99 (45b36-46a2)
It is useful at this juncture, in connection with the syntactic foundations of
syllogistic deduction, to refer back to Prior Analytics A\-6 where Aristotle estab¬
lished all the panvalid two-premiss patterns. This helps to amplify textually his
notion of formal deducibility. Respecting the two-premiss syllogism rules, from
which all extended deductive discourses are constructed, Aristotle summarized a
constituent part of his notion of deducibility five times in Prior Analytics A4~6,
in respect of each of the three figures.
Thus, it is clear when there will and when there will not be a syllogism
in this [the first] figure if the terms are universal; and it is also clear
both that if there is a syllogism, then the terms must necessarily be
related as we have said, and that if they are related in this way, then
there will be a syllogism. (A^: 26al3-16)
"Aristotle had already expressed this in Pr. An. A27: “From what had been said, then,
it is clear how every syllogism is generated, both through how many terms and premisses and
what relationships they are in to one another, and furthermore what sort of problem is proved
in each figure, and what sort in more and what in fewer figures.... For surely one ought not only
study the origin of syllogisms [43a22-23; cf. 47a2-4], but also have the power to produce them”
(43al6-24).
224
George Boger
It is evident from what has been said, then, that if there is a partial
syllogism in this [the first] figure, then it is necessary for the terms to
be related as we have said (for when they are otherwise, a syllogism
comes about in no way). ( A4 : 26b26-28)
It is evident, then, that if there is a syllogism with the terms universal
[in the second figure], then it is necessary for the terms to be related as
we said in the beginning. For if they are otherwise, a necessary result
does not come about. (A5: 27a23-25).
From what has been said, then, it is evident both that a syllogism
comes about of necessity if the terms are related to one another as
was stated, and that if there is a syllogism, then it is necessary for the
terms to be so related [in the second figure]. ( A5: 28al-3)
It is evident in this [the third] figure, then, when there will and when
there will not be a syllogism, and it is evident both that if the terms
are related as was said, then a syllogism comes about of necessity, and
that if there is a syllogism, then it is necessary for the terms to be so
related. ( A6 : 29all-14)
In each case Aristotle refers his readers to the necessary and sufficient syntac¬
tic conditions for a syllogism. He alludes to the panvalid patterns that he has
identified as deduction rules.
Perhaps the best treatment of deducibility per se in Aristotle’s logical inves¬
tigations is contained in Prior Analytics A23, a chapter some logicians believe
contains Aristotle’s attempt at a completeness proof. 100 He begins this chapter
by affirming that
every demonstration, and every deduction, must prove something ei¬
ther to belong or not to belong [avayxr) Sf) naaav atcoSei&v xai Ttavxa
auXXoytapov r] unapyov xi 7) pf) uirapyov heixvuvai], and this either
universally or partially ... either probatively or through an absurdity.
(40b23-25)
Sometimes he writes, in this connection, that every syllogism establishes one or
another problema. In any case, he then proceeds to describe the deduction process,
and his description, while not strictly a definition, amounts, nevertheless, to a kind
of stipulative definition of deducibility. We cite this passage at length to provide
his complete thinking on this matter. His syntactic treatment of the topic is
evident.
Now, if someone should have to syllogize [auXXoylaaaGca] A of B, either
as belonging or as not belonging, then it is necessary for him to take
100 Some modern logicians believe that what Aristotle writes just preceding this passage (40b23-
25) is suggestive of his interest in a completeness proof: “But it will now be evident that this
holds for every syllogism without qualification, when every one has been proved to come about
through some one of these figures” (40b20-22).
Aristotle’s Underlying Logic
225
something about something [Xot(3etv tt xaxa xivoq]. If, then, A should
be taken about B, then the initial thing will have been taken. But if A
should be taken about C, and C about nothing nor anything else about
it, nor some other thing about A, then there will be no syllogism [ouSe'u;
Ecnron ouXXoYtajJOc;] (for nothing follows of necessity [ou8ev au[2(3divEi el;
avaYxrjc;] through a single thing having been taken about one other).
Consequently, another premiss [xpoxacnv] must be taken in addition.
If, then, A is taken about something else, or something else about it or
about C, then nothing prevents there being a syllogism [auXXoYurpov],
but it will not be in relation to B through the premisses taken. Nor
when C is taken to belong to something else, that to another thing,
and this to something else, but it is not connected to B [jar) auvcixxr) 5e
TCpoc; xo B]: there will not be a syllogism [auXXoYiajioc;] in relation to
B in this way either. For, in general, we said that there cannot ever be
any syllogism [auXXoYU7[jb<;] of one thing about another without some
middle term having been taken which is related in some way to each
according to the kinds of predications [xotk; xaxrjYopicac;]. For a syllo¬
gism, without qualification, is from premisses [6 [aev yap ouXXoYicrpoe;
6 (kXw<; ex xpoxaCTEWv caxiv]; a syllogism [auXXoYiajJoc;] in relation to
this term is from premisses in relation to this term; and a syllogism of
this term in relation to that is through premisses of this term in rela¬
tion to that. And it is impossible to take a premiss in relation to B
without either predicating or rejecting anything of it, or again to get
a syllogism of A in relation to B without taking any common term,
but (only) predicating or rejecting certain things separately of each of
them. As a result, something must be taken as a middle term for both
which will connect the predications, since the syllogism [ooXXoYiapoc;]
will be of this term in relation to that. If, then, it is necessary to take
some common term in relation to both, and if this is possible in three
ways (for it is possible to do so by predicating A of C and C of B,
or by predicating C of both A and B, or by predicating both A and
B of C), and these ways are the figures stated, then it is evident that
every syllogism must come about through some one of these figures
[cpavepov cm icavxa ouXXoYUJpov avccf-xq Y^ 0 ^ 1 - &t-cc xouxwv xivot; xa>v
axrportwv]. (40b30-41al8)
Here Aristotle refers to the epistemic value of syllogisms in the deduction process.
This passage clearly indicates that a deduction of a conclusion sentence must come
from a set of sentences taken as premisses, and, moreover, that the derivation must
happen according to prescribed syntactic rules. Thus, according to Aristotle’s
formal system, there is a derivation of a categorical sentence from other given
categorical sentences, either directly or indirectly, whenever “[1] the middle is
predicated and a subject of predication, or if it is predicated and something else is
denied of it ... [or 2] if it is both predicated of something and denied of something
226
George Boger
... [or 3] if others are predicated of it, or one is denied and another is predicated”
(A32: 47bl-5).
In Posterior Analytics B4 he writes that “a deduction proves something of
something through a middle term [6 pev yap auX^oyiopoc xi xaxa xivoc; Seixvuox
Sict xou peaou]” (91al4-15; cf. Pr. An. A32 : 47b7-9). Again, consider the
following passages from Posterior Analytics B2 :
Thus it results that in all our searches we seek either if there is a
middle term or what the middle term is. For the middle term is the
explanation [acrtov], and in all cases it is the explanation that is being
sought. (90a5-7, 24)
Again, “it is plain, then, that whatever is sought, it is a search for a middle term”
(Po. An. B3: 90a35-36). And from Prior Analytics A28 we read:
This is because, in the first place, the examination is for the sake of the
middle term, and one must take something the same, not something
different. (44b38-45al)
The linchpin, then, in Aristotle’s notion of formal deducibility, a concept inde¬
pendent of an intended interpretation, albeit anticipating one for Aristotle, is his
notion of the middle term. There is no syllogism of one thing about another with¬
out taking a term in common, the sine qua non of syllogistic inference. In fact,
as we have seen (§2.3), positioning the middle term is a syntactic formation rule
of syllogistic argumentation. A careful reading of Prior Analytics and Posterior
Analytics reveals Aristotle’s preoccupation with the middle term in the deduction
process.
Finally, in Posterior Analytics A19-23, where he argues that not everything is
demonstrable and against reasoning in a circle (cf. Po. An. A3, B12 and Pr. An.
B5-7, B16), Aristotle shows that a deduction cannot contain an infinite chain of
reasoning. This position is most strongly argued in Posterior Analytics A22.
Hence if it were possible for this to go on ad infinitum, it would be pos¬
sible for there to be infinitely many middle terms between two terms.
But this is impossible if the predicates come to a stop in the upward
and the downward directions. And that they do come to a stop we
have proved generally earlier and analytically just now. (84a37-84b2)
There Aristotle shows that “one cannot survey infinitely many items in thought”
(83b6-7); “there must therefore be some term of which something is predicated
primitively, and something else of this” (83b28-29). Again, if demonstration is
possible, the predicates in between must be finite (83b38-84a6).
Thus, we see that Aristotle’s notion of formal deducibility corresponds exactly
with that of a modern logician, point for point. We can now restate a notion of
formal deducibility, this time more tailored to Aristotle’s system.
Aristotle’s Underlying Logic
227
A given categorical sentence c is formally deducible from a given set of
categorical sentences P when there exists a finite sequence of categorical
sentences that ends with c and begins with P such that each categorical
sentence in the sequence from P is either a member of P or a categorical
sentence generated from earlier sentences solely by means of stipulated
deduction rules such that the terms in c are linked through a series of
common terms from P.
Aristotle at no one place expressed this notion in just this manner, but his inten¬
tionally subscribing to such a notion, even making statements close to this effect,
is unmistakable.
5.2 Two methods of deduction in Aristotle’s system
Aristotle identified and used two methods of deduction in Prior Analytics : (1)
direct, or probative, deduction and (2) indirect, or reductio (leading to an absurdity
or per impossibile ), deduction. He makes this explicit in A23.
Now, every demonstration, and every deduction [xaaav arcoSeilfiv xai
tc&vt a auXXoyiapov], must prove [avdyxr) Seixvuvou] something either
to belong or not to belong, and this either universally or partially, and
in addition either probatively [Seixxixwp] or from an assumption [el;
OtcoGsctewc;] (for (deduction) through an absurdity [hid xou ot&ovdxou]
is a part of (deduction) from an assumption). (40b23-26)
A direct proof begins a deduction without making an assumption, and an indi¬
rect proof begins by assuming the contradictory opposite of the conclusion as an
additional premiss and then it deduces a contradiction. Aristotle understood this
in exactly the same way as a modern logician, indeed, himself having formulated
them for us in ancient times. Here follow two passages from Prior Analytics that
make this clear, the first is from A23.
For all those which come to a conclusion through an absurdity deduce
the falsehood, but prove the original thing from an assumption when
something absurd results when its contradiction is supposed [rcavxec;
yap ol 5ia too aSuvaxou Tiepdivovxet; xo pev <J*eu8o<; ouXXoylCovxai, xo
5’ e£ apxfi? UTtoGsaecx; Seixvuooaiv, oxav ahuvaxov xi ouppaivr] xfjt;
avxicpdaeax; xeGdarjp], (proving,) for example, that the diagonal is in¬
commensurable because if it is put as commensurable, then odd num¬
bers become equal to even ones. It deduces [auXXoyiCcxai] that odd
numbers become equal to even ones, then, but it proves [Seixvumv]
the diagonal to be incommensurable from an assumption since a false¬
hood results by means of its contradiction [end ijieuSoc; aupPaivei Sid
xfjv dvxlcpaaiv]. For this is what deducing [auXXoylaaaGai] through an
absurdity was: proving something impossible by means of the initial
228
George Boger
assumption [to Sei^dt xt ahuvaxov Sta xf)v uxoGeatv]. Conse¬
quently, since a probative deduction of the falsehood comes about [wax’
ETtel too (jjEuSout; ylvexoa ouXAoyiapcx; SetxxLxdc;] in those cases which
lead away to an absurdity (while the original thing is proved [Seixvuxoa]
from an assumption) ... (41a23-34; cf. Po.An. All: 77a22-25, A26:
87a6-12)
The second is from Prior Analytics B14-
A demonstration (leading) into an absurdity differs from a probative
demonstration in that it puts as a premiss what it wants to reject by
leading away into an agreed falsehood, while a probative demonstration
begins from agreed positions. More precisely, both demonstrations take
two agreed premisses, but one takes the premisses that the deduction [6
auAAoytapoc;] is from, while the other takes one of these premisses and,
as the other premiss, the contradictory of the conclusion [xfjv dvxlcpaatv
too aupnepdapaxoc;]. Also, in the former case it is not necessary for the
conclusions to be familiar or to believe in advance that it is so or not,
while in the latter case it is necessary to believe in advance that it is
not so. It makes no difference whether the conclusion is an affirmation
or a denial [cpdcnv r) dntocpacuv], but rather it is similar concerning both
kinds of conclusion. (62b29-38) 101
Aristotle, in fact, treated reductio rather fully in Prior Analytics Bll-13, treating
indirect proof in each of the three figures consecutively.
We illustrate each of the two kinds of deduction here by providing an instance
of Aristotle’s own use of each kind in the metalanguage of Prior Analytics. First,
in connection with the method of direct deduction, there is Aristotle’s text for
Camestres in the second figure.
Next, if M belongs to every N but to no X, then neither will N belong
to any X. For if M belongs to no X, neither does X belong to any M;
but M belonged to every N; therefore, X will belong to no N (for the
first figure has again come about). And since the privative converts,
neither will N belong to any X, so that there will be the same syllogism.
(It is also possible to prove these results by leading to an impossibility.)
(27a9-15)
We provide Aristotle’s text for Camestres, organized just as with Aristotle and
according to a modern sequencing on the left, and on the right we provide our
modern notation that exactly reproduces his meaning (Table 39).
101 Aristotle continues as follows: “Everything concluded probatively can also be proved through
an absurdity, and whatever is proved through an absurdity concluded probatively, through the
same terms” (62b38-41). He concludes Pr. An. B14 with this statement: “it is clear, then, that
every problem can be proved in both ways, through an absurdity as well as probatively, and that
it is not possible for one of the ways to be separated off” (63b 18-21).
Aristotle’s Underlying Logic
229
Table 39.
An instance of direct deduction
Aristotle’s text
Modern notation
1 .
M belongs to every N
1 .
MaN
2.
M belongs to no X
2.
MeX
?
neither will N belong to any X
?
NeX
3.
M belongs to no X
3.
MeX
2
repetition
4.
neither does X belong to any M
4.
XeM
2
e-conversion
5.
M belonged to every N
5.
MaN
1
repetition
6.
X will belong to no N
6.
XeN
4,5
Celarent
7.
neither will N belong to any X
7.
NeX
6
3 e-conversion
Next, we have Aristotle’s text for Baroco in the second figure, followed, again,
by a table that organizes this text according to his sequencing on the left and on
the right by our modern notation that reproduces his meaning (Table 40).
Next, if M belongs to every N but does not belong to some X, it is
necessary for N not to belong to some X. (For if it belongs to every
X and M is also predicated of every N, then it is necessary for M to
belong to every X; but it was assumed not to belong to some.) And if
M belongs to every N but not to every X, then there will be a deduction
that N does not belong to every X. (The demonstration is the same.)
(27a36-27b3)
Table 40.
An instance of indirect deduction
Aristotle’s text
Modern notation
1 .
M belongs to every N
1 .
MaN
2.
M does not belong to some X
2.
MoX
?
N not to belong to some X
?
NoX
3.
N belongs to every X
3.
NaX
assume
4.
M is also predicated of every N
4.
MaN 1
repetition
5.
M to belong to every X
5.
MaN 4,3
Barbara
6.
M not to belong to every X
6.
MoX & 2,5
conjunction &
MaX
contradiction
7.
N does not belong to every X
7.
NoX 3-6
reductio
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5.3 Aristotle’s notion of logical consequence
Aristotle defines “necessary” in Metaphysics, where he defines many philosophical
concepts, precisely as he uses the concept in Prior Analytics in relation to de¬
ciding the concludence and inconcludence of premiss-pair patterns. Our grasping
his notion of logical consequence — following from necessity, or to ic, dtv&yXT)?
ou[i(3divEtv — in Prior Analytics can be informed by what he writes in Meta¬
physics 5.5: “that which is necessary is that having no other relationship possible
[etl to (if) Ev§£)(6(iEvov aXXox; e')(£lv dvayxaiov tpapEv outgk eysiv]” (1015a33-35).
In Metaphysics 4-5 he writes much the same: “for it is not possible for what is
necessary to be one way and another way, and so if something is of necessity, it
cannot be so and not so [to yap dvayxaiov obx £v8EX£Tai aXXcoc; xai aXXax; exelv,
coot’ ei ti ecttiv e5 avayxqc;, oux eTei out<o te xai ouy outok” (1010b28-30). And,
thus, in respect of sentences, “opposed statements cannot [both] be true [at the
same time] [to (if) £ivai dXr)0£i<; apa toc; dvTLXEipsvap cpdaeu;]” (1011bl3-14).
Aristotle also affirms in this connection a principle of consistency in Prior An¬
alytics A32: “for all that is true must in all ways be in agreement with itself [8 ei
yap Tiav to aXr)0£<; auro EauTW ojioXoyoupevov iivai xavTr)]” (47a8-9). A statement
of this principle concludes On Interpretation in the following way:
It is evident also that it is not possible either for a true opinion or a true
contradictory to be contrary to another true opinion. For contraries
relate to their opposites, and concerning these it is possible to assert
truly of the same thing, but it is not possible that contraries hold of
the same thing [at the same time]. (24b6-9)
And Aristotle expressed this principle somewhat concisely in On Interpretation
14 a few passages before the one just cited: “a true opinion is never contrary to
another true opinion [ouSetcote 8e dXr)0f)<; (8o5a) aXr)0Ei SvavTta]” (23b37-38).
Returning to his notion of necessity, we find that Aristotle also makes an explicit
reference to demonstration at Metaphysics 5.5 , in connection with the passage cited
above, that conforms well with his conception of consistency:
Demonstration is of necessary things [eti f] axoSEupc; tov dvayxaiuv],
because, if there is a demonstration proper, it is not possible for there
to be any other relations [oox zvhzyzxoa dXXwg exelv]; reason for
this is the premisses [toutou 8’ diTia ra rcpcoTa], for if there is a syllogism
it is [logically] impossible for there to be another relationship among
them [el a8uvaTov aXXcoc; e^eiv e5 8>v 6 auXXoyiajJot;]. (1015b6-9) 102
Thus, a premiss-pair that results in a syllogism is such that no other result is
possible. 103 Aristotle had established in Prior Analytics A 4-6 that for a syllogism
102 Cf. Po. An. A6 on demonstration of that which is necessary. Consider that for Aristotle
scientific knowledge, apodeiktike episteme, is that which cannot be otherwise.
103 This holds notwithstanding that a weakened a or e sentence, i.e., an i or 0 sentence, is a
different sentence; this is a trivial truth for Aristotle. ‘To be otherwise’ refers to contrariety and
contradiction.
Aristotle’s Underlying Logic
231
to arise it is necessary and sufficient that the terms be related as he stated in
a number of syntax rules (§5.1). Likewise, for there not to be a syllogism, it is
necessary and sufficient that terms be related in the other ways he covered there
(§5.1). As we have seen, Aristotle established a set of formal rules, relating to
syllogistic argumentation, for deciding logical consequence.
Another indication of Aristotle’s sophistication in respect of logical consequence
concerns his discussion of the most certain principle of all, the law of non-contra¬
diction, one statement of which is expressed in Metaphysics 4-3: 1005bl9-20 (4-3-8;
cf. 11.5-6). He then writes that “someone who denies [this principle] would at the
same time hold contrary opinions. Hence, everyone who performs a demonstration
establishes it on this ultimate principle” (1005b30-33; cf. 1005b22-34). On at least
two occasions Aristotle refers to someone not subscribing to this principle as being
“no better than a plant” because nothing meaningful can be asserted and rational
discourse (logos) is destroyed (1006bo-ll, 1008b7-9).
It is evident, moreover, from his treatment of the principle of non¬
contradiction that Aristotle understands every sentence to be a logical consequence
of a contradiction. The following passage from Metaphysics 4-4 illustrates that
his grasp of logical consequence in this connection is much the same as that of
modern logicians.
If all contradictories [avxicpacreu;] were true at the same time of the same
thing, it is evident that everything would be one. For the same thing
would be a trireme, a wall, and a man, if it is possible either to affirm
or to deny something of everything [ei xaxa ncrnoi; xi r) xaxacpfjoou
r] dmocpfjaca ev8s)( £Tca ] ••• For if someone thinks that a man is not a
trireme, then clearly he is not a trireme. But if the contradictory is
[just as] true, then he is also a trireme. ... [Such persons are really
describing non-being] ... But [then] one must assert an affirmation
and a denial about every single thing. For it is absurd that the denial
holds of itself and yet excludes other denials that do not belong to it.
I mean, for example, that if it were true to say that a man is not a
man, it is evident also that he is a trireme and not a trireme. Now,
on the one hand, if the affirmation [xaxckpaac;] is admitted, it is also
necessary that the denial [omocpaau;] be admitted; on the other hand,
if the affirmation does not belong, then the denial will belong more
than the denial of itself. Now, if this denial holds, then also the denial
of the trireme will belong. And if this, also the affirmation. ... [If
this situation holds then again] any assertion whatever may be denied
and any denial whatever may be asserted [ei KEpi naoac,, roxXiv, rjxoi
xa0’ octcuv xo cpfjaai xal aKOcpqaai xai xa0’ oawv &7TOcpfjacu xod cpfjaat].
(1007bl8-1008al3)
At Metaphysics 4-5 Aristotle reiterates his thinking: “for if all opinions and ap¬
pearances are true, it is necessary that every one is true and false at the same
time ... and so it is necessary that the same thing must both be and not be [wax’
232
George Boger
dtvdyxr) to auto elvat xe xal jaf) eivoa]. And if this is so, every opinion must be
true” (1009a7-13). That every sentence follows from a contradiction is a truth for
Aristotle. 104
In addition to these discussions in Metaphysics there is a passage in Prior
Analytics B2 where Aristotle treats “following necessarily” in much the way that
modern logicians tend to deny he could if he had not defined “logical consequence”.
It is worthwhile citing this passage in its entirety, since he states his notion of
logical consequence most clearly. He writes:
First, then, it is clear from the following that it is not possible to deduce
[CTuXXoyiaaaOoa] a falsehood from truths. For if it is necessary for B to
be when A is, then when B is not it is necessary for A not to be. Thus,
if A is true, then it is necessary for B to be true, or else it will result
that the same thing both is and is not at the same time [aup(3rjaexoa
to auxo apa £ivca xe xai oux iivai]; but this is absurd. (53bll-16)
This passage continues:
But let it not be believed, because A is set out as a single term, that it is
possible for something to result of necessity [eS; otvctyxiy; xt auppchvELv]
when a single thing is, for that cannot happen: for what results of ne¬
cessity is a conclusion [to psv yap aup(3cavov iZ, otvayxr)<; to aupxepotapa
cart], and the fewest through which this comes about are three terms
and two intervals or premisses. (53bl6-20)
Aristotle explains his use of schematic letters here in the continuation of this
passage, where he addresses logical consequence in connection with syllogistic rea¬
soning by means of a syllogistic discourse. But Aristotle here immediately provides
a terse metalogical discourse on syllogistic logical consequence, but using proposi¬
tional logic. He treats a syllogism as fitting a single conditional sentence pattern.
First, he conjoins the two sentences in the premiss-set of a syllogism and takes
them as a single sentence pattern, A, which is itself the antecedent of a con¬
ditional sentence whose consequent is the conclusion pattern, B, of a syllogism:
thus, “if A then B”. He then affirms that if this is the case, then so must “if not-B
then not-A” be the case, otherwise something true would imply something false,
and this is absurd. Aristotle affirms the consistency of the following three sentence
patterns (Table 41).
Second, he affirms that “if A is true, then it is necessary for B to be true, or
else it will result that the same thing both is and is not at the same time; but
this is absurd”. We take the “or else” to refer to taking “if A is [the case] then B
is not [the case]”, or “A D -B”. Aristotle’s test of the consistency of the original
three sentence patterns is to substitute “A D -B” for “A D B” in the original set
and then to deduce a contradiction. We can represent his thinking with a familiar
notation of propositional logic as follows (Table 42).
104 However, G. Priest does not take Aristotle in Meta. 4 as treating ‘explosion’; see, e.g., G.
Priest 1998 & 2000.
Aristotle’s Underlying Logic
233
Table 41.
Aristotle’s text
Modern notation
1 .
If A is [the case], then B is [the case]
1 .
A D B
2.
A is [the case]
2.
A
3.
If B is not [the case], then A is not [the case]
3.
-B D -A
Table 42.
Original set
of sentences
Modern
notation
Test of
consistency
1. If A is [the case],
then B is [the
case]
1. A D B Given
1. A 3 -B Substitute
2. If B is not [the
case] then A is
not [the case].
2. -B D -A Given
2. -Bd -A Given
3. A is [the case].
3. A Given
3. A Given
4. -B 1,3 detachment
5. -A 2,4 detachment
6. A &; -A 3,5 conjunction,
contradiction
This seems to capture Aristotle’s thinking, which is regrettably terse. He states
that is impossible for a set of true sentences to imply a false sentence without
contradiction. His method is not that of providing a model of model sets, but one
similar to the method of reductio proof. He deduces a contradiction from a set
of sentence patterns, one of which has been substituted for one in the original set
whose consistency is to be demonstrated. The substituted sentence pattern is the
contradictory or contrary of the original sentence pattern, in this case, where ‘A
is the case’, or ‘is true’. Thus, if A is true, then for A to imply B, B must also be
true; thus, A implying not-B cannot be the case, or is false.
Aristotle continues the quotation cited above from Prior Analytics B2, and
he now represents the same notion of logical consequence, but this time with
syllogistic expressions. He writes:
But let it not be believed, because A is set out as a single term, that it
is possible for something to result of necessity when a single thing is,
for that cannot happen: for what results of necessity is a conclusion,
and the fewest through which this comes about are three terms and two
intervals or premisses. If it is true, then, that A belongs to everything
to which B belongs, and B to what C belongs, then it is necessary for
A to belong <to what C belongs to>, and this cannot be false (for the
same thing would belong and not belong at the same time). Therefore,
234
George Boger
A is put as if a single thing, the two premisses being taken together.
And similarly also in the case of privative deductions. For it is not
possible to prove a falsehood from truths [ou yap ccmv dXr)0a>v Sci^on
c|>eu§o<;]. (53bl6-25)
This, in fact, is a treatment of Barbara to illustrate the truth that true sentences
can not imply a false sentence, and, thus, it proves the logical impossibility of an
invalid argument fitting this teleios sullogismos pattern. We can illustrate rather
exactly his thinking, although tersely expressed in this passage, using his schematic
letters, his method of deduction (from Pr. An. A5-d), and interpolating the steps
in his deduction, to set it out in a familiar manner (Table 43).
Table 43.
Aristotle
’s test of consistency
Aristotle’s text
Modern notation
1. A belongs to every-
1. AaB
thing, to which B
belongs
2. B to [everything to
2. BaC
[1 k 2 = A (53bl 1—16)]
which] what C be¬
longs
? A to belong (to
? AaC
[B (53bl 1—16)]
what C belongs to)
[3-8] this cannot be false
3. AoC
Assume
(for the same thing
would belong and
not belong at the
same time)
4. AaB
1 repetition
5. AoC
3 repetition
6. BoC
4,5 Baroco
7. BaC &BoC 2,6 conj, contradiction
8. AaC
3-7 reductio
It is evident that Aristotle works with a notion of logical consequence much like
Tarski’s, with which modern logicians are familiar, namely:
A given sentence c is a logical consequence of a given set of sentences
P when every true interpretation of P is a true interpretation of c. 105
We can express Aristotle’s thinking on this matter as follows:
105 Tarski writes in “On the concept of logical consequence” (Corcoran 1990: 417): “The sen¬
tence X follows logically from the sentences of the class K if and only if every model of the class
K is also a model of the sentence X”.
Aristotle’s Underlying Logic
235
A given categorical sentence c follows necessarily from a given set of
categorical sentences P when every set of term substitutions 106 that
makes each sentence in P true makes the sentence in c true.
Now, again, Aristotle does not compose such a sentence, yet it is clear that he
formulates other statements that make his understanding clear, especially, for
example, those relating to proving inconcludence.
We can now be confident that Aristotle’s notion of “following necessarily” ex¬
presses his notion of “logical consequence”, and that he established that it is log¬
ically impossible for true sentences to imply a false sentence. Aristotle frequently
writes that it is impossible to deduce a falsehood from truths; this is the theme
of Prior Analytics B2-4- There he shows that any combination of truth-values for
premisses and conclusion can result in a valid argument except in the case of an
argument with all true premisses and a false conclusion. In Prior Analytics B2 he
explicitly writes:
Now, it is possible for circumstances to be such that the premisses by
means of which the syllogism comes about are true, or that they are
false, or that one premiss is true and the other false. The conclusion,
however, is either true or false of necessity. It is not possible, then,
to deduce [auXXoyloaaGai] a falsehood from true premisses, but it is
possible to deduce a truth from false ones (except that it is not a
deduction of the ‘why’ but of the ‘that’, for a deduction [auXXoyi.a[i6<;]
of the ‘why’ is not possible from false premisses). (53b4-10)
This translates into affirming that no argument is valid that has true premiss
sentences and a false conclusion sentence. Aristotle then follows this passage to
provide his reasoning for this condition, which we have treated just above ( Pr.
An. B2: 53bll-25).
6 SUMMARY OF ARISTOTLE’S ACCOMPLISHMENTS IN PRIOR
ANALYTICS
By the end of Prior Analytics A6 Aristotle had systematically worked through all
possible patterns of two categorical sentences that could serve in the role of pre¬
misses to discover “how every syllogism comes about”. He established that among
these possible patterns there are 14 that result, in their application to an object
language, in something following necessarily, that is, that result in syllogisms. To
accomplish this project, Aristotle invented a formal language to devise a rudimen¬
tary model of his logic in Prior Analytic. In this way he was able to describe a
deduction system and demonstrate certain logical relationships among syllogistic
rules, not the least of which accomplishment was establishing the independence
106 While substitution and reinterpretation are distinct logical concepts, their application
amounts to the same thing.
236
George Boger
of a small set of deduction rules (A 7). The formal language used in Prior An¬
alytics built upon the foundation of his linguistic studies in Categories and On
Interpretation. Strictly speaking, Aristotle’s formal language does not consist in
sentences, as ‘sentence’ is defined in On Interpretation and as ‘ protasis' is used in
Prior Analytics. Rather, his formal language consists in relatively uninterpreted
sentence patterns. By substituting non-logical constants — a predicate term and a
subject term — for schematic letters, Aristotle could produce any number of object
language sentences. We could easily call such sentences interpretations without
distortion, as a modern logician understands this notion. This, however, would
misrepresent Aristotle’s logic. Nevertheless, this closeness to modern practice is
not a superficial resemblance, but an indication of Aristotle’s genius and original¬
ity. Here we summarize some of his accomplishments and insights into logic with a
synopsis of his model (§6-1) and with a summary of four proof-theoretic processes
he employed in Prior Analytics (§6.2).
6.1 Synopsis of Aristotle’s model
Aristotle invented his formal language with an aim to model scientific discourse.
Such discourse, then, might be taken as its ‘intended interpretation’. In any case,
using a modern mathematical template, we can re-present Aristotle’s own model
in the following way (Table 44).
6.2 Four proof-theoretic processes in Prior Analytics
Aristotle did not describe deductions in Prior Analytics A4-6 but showed how ev¬
ery syllogism comes about. He also explained how syllogisms do not come about
and he refined his system. He described a natural deduction system of an underly¬
ing logic in ways suggestive of modern methods, and he proved certain properties
of this deduction system. His treatment of his logic is thoroughly metalogical.
Here we briefly summarize four proof-theoretic processes used in Prior Analytics.
All four processes have their counterparts in one or another object language.
Deciding concludence: the method of completion
Completion ( teleiosis , teleiousthai) is a proof-theoretic, deductive process that
establishes knowledge that a given argument pattern with places for two premisses
having places for three different terms is panvalid by using the patterns of the
teleioi sullogismoi as deduction rules. Completion is an epistemic process. In
Prior Analytics Af-6 Aristotle established the preeminence of the patterns of
the teleioi sullogismoi among the panvalid patterns or, conversely, he implicitly
established that the patterns of the ateleis sullogismoi are redundant rules in
his deduction system. The process of completion per se does not establish that
any rule of deduction is redundant. Nor does completion involve transforming a
given argument pattern into another argument pattern, since in the process of
deduction a given argument pattern is not itself transformed but shown to be
Aristotle’s Underlying Logic
237
panvalid through a chain of reasoning cogent in context, which chain of reasoning
is generated by means of specified deduction rules. Aristotle’s metalogical theorem
concerning completion is that “all the ateleis sullogismoi are completed by means
of the first figure syllogisms using probative and reductio proofs” (A 7: 29a30-
33). Aristotle reserved using the verb ‘ teleiousthaV specifically in relation to a
process by which a valid argument, whose validity is not evident, is made evident
by performing a deduction during which a teleios sullogismos , one whose validity
is obviously evident, is generated; this signals cogency in the deduction process
from premisses to conclusion.
Deciding inconcludence: the method of contrasted instances
The method of contrasted instances used in Prior Analytics Af-6 is the comple¬
ment of the process of completion. The purpose of this method is to establish which
elementary argument patterns are not panvalid. This proof-theoretic method is
different from the method of counterargument, since (1) it treats patterns of pre¬
misses and argument patterns and not arguments and, thus, it establishes panin¬
validity and not invalidity, and (2) it does not produce an argument in the same
form as a given argument but with true premisses and a false conclusion. Rather,
this method constructs two arguments, each of whose premisses are true sentences
fitting the same premiss-pair pattern and whose conclusions also are true sen¬
tences, but in the one argument the conclusion is an a sentence, in the other an
e sentence. This establishes that a given premiss pair pattern is inconcludent and
that consequently its corresponding four argument patterns are paninvalid. No
syllogism is possible in such a case. It is not possible to construct such arguments
with a concludent premiss pair pattern: in that case every similar construction
that produces true sentences as premisses results in at least one false sentence
among the conclusions. Thus, any two sentences of three terms fitting a given
inconcludent premiss-pair pattern are shown never to result together in a valid ar¬
gument. In this way Aristotle was able to eliminate would-be syllogistic deduction
rules.
238
George Boger
Table 44.
An ancient model of an underlying logic
Aristotle’s own model
Aristotle’s model
expressed by a
modern notation
LANGUAGE
LANGUAGE
Vocabulary
Vocabulary
1 .
Four fully interpreted logical
constants
1 .
Logical constants
belongs to every
a
belongs to no
e
belongs to some
i
belongs not to every
0
2.
n schematic (upper case) let¬
ters intended to hold places for
non-logical constants (terms)
2.
n schematic letters
A, B, C; M, N, X; P, R,
S...
A, B, C...
Grammar
Grammar
1 .
Sentences are the elements of
1 .
Categorical sentence
a language. A categorical sen¬
tence is formed by concatenat-
patterns
ing a non-logical constant with
a logical constant with a differ¬
ent non-logical constant.
A belongs to every B.
AaB
A belongs to no B.
AeB
A belongs to some B
AiB
A belongs not to every B.
AoB
2.
Relationships of opposite sen¬
tences
2.
Sentential relationships
Contradictories
Contradictories
A belongs to every B — to —
A belongs not to every B.
AaB — to — AoB
A belongs to no B — to — A
belongs to some B
AeB — to — A«B
Contraries
Contraries
A belongs to every B — to —
A belongs to no B.
AaB — to — AeB
Aristotle’s Underlying Logic
239
3. Premiss formation
3. Premiss formation
One-premiss argument
One-premiss argument
Take any one of the four cate-
gorical sentences
AB
AtB
Two-premiss argument
Two-premiss argument
Take any two of the four cat-
egorical sentences with three
different terms, one in common
First figure:PMS
First figure:PMS
1. PM
1. PxM
2. MS
2. MyS
Second figure:MPS
Second figure:MPS
1. MP
1. MxP
2. MS
2. MyS
Third figure: PSM
Third figure: PSM
1. PM
1. PxM
2. SM
2. SyM
4. P-c argument formation
4. P-c argument formation
One-premiss (conversion) ar-
One-premiss argument
gument
1. AB
1. AxB
?. BA
?. ByA
Two-premiss argument
Two-premiss argument
First figure:PMS
First figure:PMS
1. PM
1. PxM
2. MS
2. MyS
?. PS
?. PzS
Second figure:MPS
Second figure: MPS
1. MP
1. MxP
2. MS
2. MyS
?. PS
?. PzS
Third figure: PSM
Third figure: PSM
1. PM
1. ParM
2. SM
2. SyM
?. PS
?. P.zS
240
George Boger
DEDUCTION SYSTEM
DEDUCTION SYSTEM
1. Deduction rules
1. Deduction rules
One-premiss conversion rules
One-premiss rules
If A belongs to every B, then
1. AoB 1. Az'B 1. AeB
B belongs to some A.
If A belongs to some B, then
B belongs to some A.
If A belongs to no B, then B
belongs to no A.
BiA BzA BeA
Two-premiss syllogism rules
(reduced system)
Two-premiss rules
If A belongs to every B and
B belongs to every C, then A
belongs to every C.
1. AaB 1. AeB
If A belongs to no B and B be¬
longs to every C, then A be-
2. BaC 2. BaC
longs to no C.
.-. AaC AeC
2. Types of deduction
2. Types of deduction
Direct deduction
Direct deduction
P
Step 1
[See section 5.2]
step n — 1
step n = c
Indirect deduction
Indirect deduction
P
Step 1: contra of c
[See Section 5.2]
step n — 1 : X [corij & contr]
step n = c
Aristotle’s Underlying Logic
241
SEMANTICS
1. Meanings of sentences
AaB: universal attributive:
Every B has property A
AeB: universal privative: No B
has property A.
AiB: partial attributive: Some
B has property A
AoB: partial privative: Some B
does not have property A.
2. Truth-values of sentences
AaB is true iff every B has
property A.
AeB is true iff no B has prop¬
erty A.
Aib is true iff some B has prop¬
erty A.
AoB is true iff some B does not
have property A.
3. Logical Consequence
It is impossible that a false sen¬
tence follows necessarily from
true sentences.
SEMANTICS
1. Meanings of sentences
[Same]
2. Truth-values of sentences
[Same]
3. Logical consequence
[Same]
Transforming patterns: analysis
Analysis ( analusis, analuein) is a proof-theoretic process that transforms one syl¬
logistic pattern in any one figure into another syllogistic pattern of another figure
only if both patterns ‘prove’ the same problema. Analyses are performed through
conversion and premiss transposition. Analysis is not directly concerned with
making validity or panvalidity evident, not with a deduction process, nor with es¬
tablishing whether a given syllogistic pattern is a redundant rule. Rather, Aristotle
aimed to promote his students’ facility with reasoning syllogistically to establish
(to xaxaoxeudCEtv) and to refute (to otvaoxeudCexv) arguments by studying the
logical relationships among their patterns. 107 This is analogous to how modern
logicians have studied the relationships among the rules of propositional logic.
Aristotle’s theorem concerning analysis is that ‘the syllogisms in the different fig¬
ures that prove the same problema are analyzable into each’ (see Af5: 50b5-7).
107 See Pr. An. A26-28 and summary at A30: 46a3-10. For example, Aristotle writes (A26):
"... a universal positive problema is most difficult to establish [xaxaaxeudoai] but easiest to
refute [dvacrxeudaai]” (43al-2). Cf. Aristotle’s projects in writing SR and Top.
242
George Boger
7 CONCLUDING REMARKS
We have represented Aristotle as having modelled his syllogistic as an underlying
logic according to the practice of a modern mathematical logician. In Prior An¬
alytics he especially articulated the logical syntax of his syllogistic system while,
nevertheless, always presupposing its applicability to various axiomatic sciences.
Aristotle conceived of his system as a formal calculus, akin to mathematical cal¬
culi, since his aim was to establish a reliable deduction instrument for episteme
apodeiktike. Some modern logicians believe that, since Aristotle did not specif¬
ically refer to alternative interpretations or perform operations that suggest his
recognizing this, he must have taken his ideal language to be fully interpreted. In
this connection, then, they believe that Aristotle could not have conceived of a
language apart from its intended interpretation, that Aristotle did not distinguish
logical syntax from semantics. However, when we consider Aristotle’s accomplish¬
ments in Prior Analytics along with his other accomplishments in Categories , On
Interpretation , and Metaphysics , this interpretation seems not to accord with his
having invented a formal language.
One way sufficient for determining whether or not a logician distinguishes logi¬
cal syntax from semantics is to ascertain whether a logician works with notions of
interpretation and reinterpretation. In a reinterpretation one leaves the language
fixed but changes its meaning. It is thought that Aristotle’s having notions of
interpretation and reinterpretation was precluded by his not having distinguished
syntax and semantics in his logic. Perhaps, however, it is rather that his dis¬
tinguishing syntax and semantics is thought to have been precluded by his not
having notions of interpretation and reinterpretation because he did not work
with model-theoretic and set-theoretic notions. In this connection, then, we can
recognize that another equally sufficient way for determining whether or not a logi¬
cian distinguishes logical syntax from semantics is to ascertain whether a logician
works with a notion of substitution. In a substitution one changes the language, or
the content words and phrases in a given language, while leaving their meanings
and the logical form fixed. While we might agree that Aristotle did not have nor
work with fully modern notions of interpretation and reinterpretation per se, he
has nevertheless quite ably distinguished syntax and semantics as is evidenced by
his inventing and using a formal language that contains only sentence patterns.
And we have already witnessed an instance of Aristotle experimenting with rein¬
terpreting a word in much the same way as a modern logician. While substitution
and reinterpretation are distinct logical concepts, their application amounts to the
same thing. In this light, observing Aristotle’s pervasive use of schematic letters
and his common practice of substitution for establishing inconcludence, we rec¬
ognize his making a more determinate distinction between semantics and syntax
than previous interpreters have allowed. By substituting terms for schematic let¬
ters, Aristotle was able to produce an unlimited number of sentences according
to his definition in On Interpretation and his formal grammar in Prior Analytics.
This method of producing sentences from patterns surely amounts to ‘giving an
Aristotle’s Underlying Logic
243
interpretation’, while not itself, of course, strictly an interpretation. Moreover,
there are much the same results in relation to recognizing the underlying struc¬
tures of natural languages and logical languages. Again, he might easily have
construed these as interpretations of his formal language. Thus, we believe that
there are sufficient textual grounds for imputing to Aristotle a belief not only in
argument ‘forms’, but also, then, in distinguishing syntax and semantics, indeed,
in a way familiar to A. Church, A. Tarski, and other modern logicians. While it is
doubtful that Aristotle had a modern theory of language, and surely not himself
a string-theorist, it is nevertheless evident that he recognized different patterns
to underlie sentences involving, for example, ambiguity and equivocation. Indeed,
identifying these forms or patterns and establishing their logical relationships were
precisely the focus of his project in Prior Analytics A 4 - 6 , 7, and 45-
Aristotle’s notion of substitution, then, was sufficiently strong for his distin¬
guishing logical syntax and semantics. In this connection he was also able to
distinguish validity from deducibility sufficiently to note the completeness of his
logic in Prior Analytics A30. Consider the following passage:
Consequently, if the facts concerning any subject have been grasped,
we are already prepared to bring the demonstrations readily to light.
For if nothing that truly belongs to the subjects has been left out of our
collection of facts, then concerning every fact, if a demonstration for it
exists, we will be able to find that demonstration and demonstrate it,
while if it does not naturally have a demonstration, we will be able to
make that evident. (46a22-27)
This statement surely indicates that he believed his deduction system sufficiently
strong to deduce every logical consequence of a given set of sentences. And always
underpinning his thinking lay his taking such sentences to be the first principles of
axiomatizable sciences and his aspiration that his deductive sciences would be uni¬
versally complete. In Prior Analytics Aristotle turned his attention toward objec¬
tifying the formal deduction apparatus used to establish scientific theorems. Since
the process of deduction is topic neutral and formal, Aristotle was concerned with
matters of syntax and deducibility: he treated these matters especially in Prior
Analytics A. Since Aristotle was concerned with logical consequence and truth, he
was occupied also with semantic matters: he treated these matters especially in
Prior Analytics B among the other places we have examined.
ACKNOWLEDGEMENTS
Some parts of this study were treated earlier in Boger [1998] and Boger [2001]. I
wish especially to express my sincere gratitude to Canisius College for granting
me a Sabbatical Leave for Fall 2000 that enabled me to accomplish this study.
Professor John Corcoran of the University at Buffalo provided invaluable critical
comments and encouragement.
244
George Boger
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ARISTOTLE’S MODAL SYLLOGISMS
Fred Johnson
Considering Aristotle’s discussion of syllogisms as a whole, the most striking point is
that its focus is the modal syllogisms - This is the point on which the logical tradition
has diverged most completely from Aristotle, as a rule giving no attention to modal
syllogisms .... Paul Henle
Aristotle’s system of modal syllogisms, to be found in chapters 3 and 8-22 of the first
book of the Prior Analytics, has been open to public inspection for over 2300 years.
And yet perhaps no other piece of philosophical writing has had such consistently bad
reviews. Storrs McCall
... by raising the [completeness] problem, Aristotle earns the right to be considered
not only the father of logic, but also the (grand)father of meta- logic. Jonathan Lear
Storrs McCall [ 1963] developed the first formal system, the L-X-M calculus, for which
a decision procedure for assertion or rejection of formal sentences is given that has any
chance of matching Aristotle’s judgments about which of the n-premised (for n > 2)
“apodeictic syllogisms” are valid or invalid. McCall’s remarkable results were achieved
by extending Jan Lukasiewicz’s [1957] decision procedure for assertion or rejection of
expressions in his formal system, LA, that is designed to capture Aristotle’s judgments
about which of the “assertoric (or plain 1 ) syllogisms” are valid or invalid.
Lukasiewicz also considers using his four-valued modal system, the L M system, to
present Aristotle's syllogistic but finds that the match is not very good. Peter Geach also
proposes a system for dealing with the apodeictics. But, again, the match is not very
good. After examining McCall’s L-X-M system and work related to it we shall turn to his
work on the “contingent syllogisms”. His purely syntactic system, Q-L-X-M, has some
unAristotelian features that lead us to develop a modified system, QLXM'. A semantics
for QLXM' is developed that enables us to provide formal countermodels for a large
percentage of the assertoric, apodeictic or contingent syllogisms that Aristotle explicitly
considered to be invalid.
1 LUKASIEWICZ’S ASSERTORIC SYSTEM, LA
For Lukasiewicz, Aristotle’s syllogisms are “implicational” rather than “inferential”. He
says in [1957, p. 21]:
'in [1964] P T. Geach prefers ‘plain’ over ‘assertoric’.
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
© 2004 Elsevier BV. All rights reserved.
248
Fred Johnson
Syllogisms of the form:
All B is A;
all C is B;
therefore
all C is A
are not Aristotelian. We do not meet them until Alexander. This transference
of the Aristotelian syllogisms from the implicational form into the inferential
is probably due to the Stoics.
So, Lukasiewicz claims Aristotle construed the above syllogism, with traditional name
‘Barbara’, as a conditional claim:
If all B are A then if all C are B then all C are A.
Robin Smith’s [1989, p.4] translation of Barbara at Prior Analytics 25b37-40 seems to
conform with Lukasiewicz’s view:
... if A is predicated of every B and B of every C, it is necessary for A to be
predicated of every C ....
But see [Corcoran, 1972] and [Smiley, 1973] for the view that Aristotle developed natural
deduction systems rather than the axiomatic systems of the sort Lukasiewicz envisages.
Lukasiewicz uses Polish notation, a parenthesis-free notation, to express the well-
formed formulas (wffs) in his formal system, which we refer to as LA. We replace his
notation with current “standard” notation when giving the basis for it. 2 So, for example,
his Cpq (‘If p then q’) is our (p -4 q). His Np (‘not p’) and Kpq (‘p and q') are our
and (p A q), respectively.
Lukasiewicz’s assertions and rejections are marked by h and H , respectively. The sys¬
tem that is essentially Lukasiewicz’s will be called LA.
So, for example, h (Aba -4 (Acb -4 Aca)) says that Barbara is asserted in LA, which
is true. H A6a says that Aba is rejected in LA, which is true. Assertions and rejections are
relative to systems. We shall avoid using h LA , say, and rely on the context to indicate
that the assertion is relative to system LA.
Primitive symbols
term variables
monadic operator
dyadic operator
quantifiers
parentheses
a, b, c,... (with or without subscripts)
A, I
(,)
2 The manner of presentation of this system is heavily influenced by Hughes and Cresswell's presentations
of various systems in [19961.
Aristotle’s Modal Syllogisms
249
Formation rules
FR1 If Q u is a quantifier and x and y are term variables then Q u xy is a wff.
FR2 If p and q are wffs then -<p and (p —» q) are wffs .
FR3 The only wffs are those in virtue of FR1 and FR2.
So, for example, Aab, lab and (Abe —» -T6c) are wffs. Read them as ‘All a are b\
‘Some a are b’ and ‘If all b are c then it is not true that some b are c\ respectively.
Definitions
Def A (p A q) = df ->(p -4- ~^q)
Def <-> (p<r> q) = df ((p -> q) A (q 4- p))
DefE Exy — d f ->Ixy
Def O Oxy = d f -'Axy
Eab and Oab may be read as ‘No a are b’ and ‘Some a are not b\ respectively.
Lukasiewicz’s LA contains theses that are “assertions” (indicated by h ) as well as the¬
ses that are “rejections”(indicated by H ). We begin with the former, which are generated
by assertion axioms and assertion rules.
Assertion axioms
AO (PC).
A1
A2
A3 (Barbara)
A4 (Datisi)
If p is a wff that is valid in virtue of the propositional calculus (PC) then
h p (that is, p is asserted). (So, for example, h (Aab —> Aab) since it is not
possible that the antecedent Aab is true and the consequent Aab is false.
And h ((Aab -» lab) -» (->lab -> -*Aab)) since it is not possible that all
of these conditions are met: (Aab —> lab) is true, -<Iab is true and ->Aab
is false.)
h Aaa
h I aa
h (Abc —> (Aab —» Aac))
h (Abe —> (Iba —> lac))
Transformation rules for assertions
ArI (Uniform substitution for assertions, US) From h p infer h q (that is, from the as¬
sertion of p infer the assertion of q) provided q is obtained from p by uniformly
substituting variables for variables. (So, for example, from h (Aab —>■ Iba) we may
infer h (Acb -¥ Ibc) and i ~(Abb —> Ibb), by rule US. But rule US does not permit us
to infer that h (Aa &—> Iba) given that —>■ Ibb).
Ar 2 (Modus Ponens, MP) From h (p -» q) and h p infer h q.
Ar3 ( Definiens and definiendum interchange for assertions, DDI) From
h (... a ...) and a = d f P infer h (.../?...), and vice versa. (So, for example, from
h (-i/a6 —» ~<Aab) infer h (Eab -> Oab) by two uses of DDI, given definitions Def
E and Def O. Typically a use of DDI will be indicated by simply referring to a def¬
inition that is used. So, from (-■lab —> Aab) infer H (Eab —> -> Aab ) by DefE. It
is to be understood that DDI is also used.)
250
Fred Johnson
Given the assertion portion of the basis for LA, we shall give some “assertion deduc¬
tions” — sequences of wffs such that each member of the sequence is either an assertion
axiom or is entered from a prior member of the sequence by using a transformation rule
for assertions — that capture some Aristotelian principles involving conversions, subor¬
dinations, and oppositions.
Theorem 1.1. (Assertoric conversions,Con) h (Iab —> Iba) and h (Eab —> Eba).
Proof.
1. h (Abc —> (Iba -4 lac)) (by A4)
2. *" (Abb —> (Iba -4 lab)) (from 1 by US)
3. h Abb (by A1 and US)
4. h (Iba —> lab) (from 2 and 3 by MP)
5. h (Iab — » Iba) (from 4 by US)
6 . h ((Iba —> lab) -4 ( ~>Iab —> -<Iba)) (by AO)
7. h (^Iab — > ->Iba) (from 6 and 4 by MP)
8. h (Eab —> Eba) (from 7 by Def E, using DDI) ■
The above reasoning may be presented more succinctly by using the following derived
rule for assertions.
DR1 (Reversal, RV) i) From h (p -4 q) infer t ~(-'9 -4 ~>p ); ii) from h (p -4 (q -4 r))
infer h (p -4 (->r -4 -> 9 )); and iii) from H (p -4 (9 -4 r)) infer k (t -4 (p -4
-><?))•
Proof, i) Suppose h (p -4 9 ). By AO h ((p -4 9 ) -4 (-19 -4 ->p)). By MP h (->9 -4 ->p).
ii) Suppose h (p — » (9 —> r)). By AO h ((p —> (9 -4 r)) -4 (p -4 (-ir -4 9))). By MP
H (p ~^ C -17- ~ , 9 )))- Use similar reasoning for iii). ■
So, the annotation for line 7 in the above deduction may read: ‘(from 4 by RV)’. Line
6 may be deleted.
The following derived rules are useful in generating other principles.
DR2 (Assertion by antecedent interchange, AI) From h (p -4 (9 — ¥ r)) infer h (q -4
(P -f r)).
Proof. Assume h (p -4 ( 9-4 r)). By AO h ((p —> (9 -4 r)) —> (9 -4 (p -4 r))). By MP
h ( 9 -> (p-4 r)). ■
DR3 (Assertion by antecedent strengthening (or equivalence), AS) From h (p — Kq -4r))
and h (s —>q) infer h (p -4(s — >r )); and from h (p —Kq -4r)) and h (s -4p) infer h (s
-*(q ^r )). 3
3 ‘Cut’ is also used to refer to these rules.
Aristotle's Modal Syllogisms
251
DR4 (Assertion by consequent weakening (or equivalence), CW) From h (p -4q) and
h (q -4r) infer h (p -4r); and from h (p-f(q -4r)) and h (r -4s) infer h (p -4(q -4s )). 4
To prove DR3 and DR4 use AO and MP.
Theorem 1.2. (Assertoric subalternations, Sub-a) i) h (Aab -4 lab); and ii) h (Eab —»
Oab).
Proof.
1. h (Abe -4 ( Iba -4 lac)) (by A4)
2. *"( Iba -A ( Abe —> lac)) (from 1 by AI)
3. h (/aa -4 ( Aac -4 lac)) (from 2 by US)
4. h Iaa (by A2)
5. h (Aac -4 lac) (from 3 and 4 by MP)
6. h (Aab -4 lab) (i, from 5 by US)
7. h (-i lab -> -i Aab) (from 6 by RV)
8. h (Eab -4 Oab) (ii) from 7 by DDI, using DefE and Def 0) ■
Theorem 1.3. (Assertoric conversion per accidens , Con(pa)) i) h (Aab —> Iba); and ii)
h ( Eab -4 Oba). 5
Proof.
1. (Aab -4 lab) (by Sub-a)
2. (lab -4 Iba) (by Con)
3. (Aab -4 Iba) (i, from 1 and 2 by CW)
4. (Eab -4 Eba) (by Con)
5. (Eba —» 06a) (from 4 by Sub-a and US)
6. (Eab -4 Oba) (ii, from 4 and 5 by CW) ■
The following derived rule, proven by using AO and MP, is useful in proving the next
theorem.
DR5 (Biconditional rule, BIC) From h (p -4 q) and h (q -4 p) infer h (p <4 q). 6
Proof. Suppose h (p — » q) and h (q —> p). By AO, h ((p —> q) —> ((q -4 p) -4 (p <4 q))).
By two uses of MP, h (p 54 </). ■
Theorem 1.4. (Assertoric oppositions, Opp) i) h (-c4ab <4 Oab); ii) h (-^Eab <4 Tab);
iii) (-i/ab <4 Eab); and iv) (~<Oab o Aab).
4 ‘Transitivity’ and ‘Hypothetical syllogism' are also used to refer to the first of these two rules.
5 [. M. Bochenski, on p. 212 of [1963], states that the “law of accidental conversion of the universal negative
is not in Aristotle”. He is not saying that Aristotle considered inference ii) to be invalid.
6 This rule is discussed, but not named, on p. 29 of [Hughes and Cresswell, 1996],
252
Fred Johnson
Proof
1. h (-iAab —4 -> Aab ) (by AO)
2. H (->Aab -4 Oab ) (from 1 by Def O)
3. H (Oab —4 ->Aab) (from 1 by Def O)
4. h (-*Aab 44 Oab) (i, from 2 and 3 by BIC)
5. h (->Eab —4 -i Eab) (by AO)
6. H (-.Eafe -4 -r^Iab) (by Def E)
7. h (-i-i/a6 -4 Ja6) (by AO)
8. h (->Eab —4 lab) (from 6 and 7 by CW)
9. h (-i-i lab -4 -iEab) (from 5 by Def E)
10. h (/ab-4 —fob) (by AO)
11. h (lab — 4 -> Eab ) (from 10 and 9 by CW)
12. h (-i£a6 /ab) (ii, from 11 by BIC)
13. h ((-iAab 44 Oab) -4 (~<Oab 44 Aab)) (by AO)
14. h (^Oab 44 Aab) (iv, from 4 and 13 by MP)
15. h ((-*Eab 44 lab) -4 (-Uab 44 Eab)) (by AO)
16. h (-i/ab 44 Eab) (iii, from 12 and 15 by MP) ■
The following derived rule is useful in conjunction with the assertoric oppositions.
DR6 (Substitution of equivalents, SE) From h (p 44 q) and h (. . .p. ..) infer h (... q ...).
Proof. Use mathematical induction. ■
So, for example, from h (Aab —4 (Abe -4 (->Aad —4 -iAcd))) infer h (Aab —4 (Abe —4
(Oad -4 Ocd))) by SE, given the oppositions Opp.
On table 1 assertions corresponding to the familiar two-premised syllogisms are listed.
In the right column a method of deducing the assertion is given. So, for example, Barbara
is trivially asserted by using axiom A3. Celarent is asserted since the assertion of 11
(Disamis) may be transformed into h (-i lac —4 (Aba —4 -Tbc)) (by RV), which may
be transformed into h (Eac —4 (Aba —4 Ebc)) (by SE, since h (Eac 44 —>Iac) and
h (Ebc 44 —>T6c)), which may be transformed into 2 (by US, putting ‘b' in place of ‘a’
and ‘a’ in place of ‘b’). Darii is asserted since the assertion of 12 may be transformed into
h (Abc -4 (lab -4 lac)) (by AS, since (lab -4 Iba)).
1.1 Rejection in LA
Lukasiewicz uses the notion of “rejection” to develop his formal system. 7 He shows that
the invalid syllogistic forms expressed by “elementary wffs” may be rejected by augment¬
ing his formal system for assertions by adding one rejection axiom and four transforma¬
tion rules that generate rejections. We shall illustrate this claim but not give a full account
7 Smiley, in his influential article [1996], points out that Carnap and Lukasiewicz were the first logicians to
formalize the notion of rejection. Smiley attributes the shunning of rejection by most logicians to Frege’s [I960].
Smiley effectively argues that Frege’s rejection of rejection, using Occam’s razor, was unfortunate, and Smiley
shows how rejection may be put to good use in ways other than those envisioned by Camap or Lukasiewicz. For
recent work on rejection that is stimulated by Smiley’s article see [Rumfitt, 1997] and [Johnson, 1999b],
Aristotle's Modal Syllogisms
253
Table 1. Deductions in system LA
Figure 1
Barbara (1)
Celarent (2)
Darii (3)
Ferio (4)
h (Abc -4 ( Aab -4 Aacj)
h (Ebc -4 ( Aab — i Eac))
h (Abc -4 (lab -4 lac))
h (Ebc -4 (lab -4 Oac))
A3
11 ,RV,SE,US
12, AS
12,RV,SE,US
Figure 2
Cesare (5)
Camestres (6)
Festino (7)
Baroco (8)
h (Ecb -4 ( Aab — » Eac))
^ (Acb -A (Oaft — > Eac))
h (Ecb -4 (/aft -4 Oac))
h (Acft -4 (Oaft — > Oac))
12,RV,SE,AI,US
3,RV,SE,US
11,RV,SE,AI,US
1,RV,SE,US
Figure 3
Darapti (9)
Felapton (10)
Disamis (11)
Datisi (12)
Bocardo (13)
Ferison (14)
h (Aftc —t (Afta —> /ac))
^(Ebc —> (Afta -4 Oac))
h (Ibc —> (Afoa —» /ac))
h (A6c -4 (/fta — /ac))
h (Oftc — > (Afta -4 Oac))
h (Ebc —> (/fta — t Oac))
12,AS
20,RV,SE,US
12,AI,US,CW
A4
1,RV,SE,US
3,RV,SE,US
Figure 4
Bramantip (15)
Camenes (16)
Dimaris (17)
Fresison (18)
Fesapo (19)
DnAori nm
h (Acb — i (Aba -A /ac))
h (Acft -4 (iJfta -4 £ac))
h (/cft-4 (Aba ^ lac))
h (Ecb -4 (/fta -> Oac))
h (£cft -4 (Afta -4 Oac))
( A _A ^ /I n V. T r-t \
20,AI,US,CW
17,RV,SE,AI,US
3,AI,US,CW
17,RV,SE,AI,US
15,RV,SE,AI,US
i nxu
Celaront (21)
Cesaro (22)
Camestrop (23)
Camenop (24)
h (Ebc -4 (Aab —> Oac))
h (Ecb-> (Aab -4 Oac))
h (Acb -4 (Eab -4 Oac))
h (Acb -4 (Eba^Oac))
9,RV,SE,US
9,RV,SE,AI,US
20,RV,SE,US
15,RV,SE,AI,US
of Lukasiewicz’s work on rejections, which would require showing that all wffs may be
“reduced” to sets of elementary wffs.
Definition 1.5. (elementary wff and simple wff) x is an elementary wff iff x has form
(x\ -> (X 2 -> (X 3 . x n ).. .), where each Xi is a simple wjf, a wff of form Apq, Ipq,
Opq or Epq.
Rejection axioms for LA
R1 ~*(Acb -4 ( Aab —> lac))
Rejection transformation rules for LA
Rr 1 (Rejection by uniform substitution, R-US) If and x is obtained from y by uniform
substitution of terms for terms, then “ l t/.
Rr 2 (Rejection by detachment (or Modus Tollens), R-D) From h (a; -4 y) and H r/ infer
254
Fred Johnson
Rr3 (Slupecki’s rejection rule, R-S) From H (x —>z) and H (y —rz) infer
H (x -4(y —>z)) provided: i) x and y have form -> Apq or ->7pg; and ii) 2 has form
(xi —> (X 2 -4 (X 3 —> ... x n )...) where each X{ is a simple sentence.
Rr4 ( Definiens and definiendum interchange for rejections, R-DDI) From
ct...) and a =# (3 infer “ l (.../?...), and vice versa. (So, for example, from
H (-’Aa& —4 -i lab ) infer H (Eab —¥ Oab) by two uses of R-DDI, given definitions
Def O and Def E.)
The following derived rules for rejections, which are counterparts of derived rules for
assertions, are useful in simplifying presentations of rejection deductions — sequences of
wffs in which each member of the sequence is either an (assertion or rejection) axiom or
is entered by an (assertion or rejection) transformation rule, where the last member of the
sequence is a rejection.
R-DR1 (Rejection by reversal, R-RV) i) From H (p —4 q) infer H (->g —4 ->p); ii) from
~*(P -> {q —■> r*)) infer H (p —4 (~>r -4 ->g)); and iii) from ~ l (p —4 (q -4 r)) infer
H (-nr -4 (p -4 ->q)).
Proof, i) Suppose H (p -4 q). By AO (or PC) h ((->g -4 ->p) -4 (p -4 9 )). By R-D
H (-ig -4 - 75 ). ii) Suppose H (p — 4 (7 —4 r)). By AO h ((p -4 (~>r —4 ->g)) —4 (p —4 (7 —4
r))). By R-D H (p -4 (-nr -4 ->g)). Use similar reasoning for iii). ■
R-DR2 (Rejection by antecedent interchange, R-AI) From H (p -4 (q -4 r)) infer
H (9 -> (P -> 0).
Proof. Assume H (p —4 (q —4 r)). By AO h ((g -4 (p -4 r)) —4 (p -4 (7 —4 r))). By R-D
H (? (p -> f)). ■
R-DR3 (Rejection by antecedent weakening (or equivalence), R-AW) i) From H (p —4
(q —4 r)) and h (g —4 s) infer H (p -4 (s —4 r)) ; and ii) from H (p -4 (g —4 r))
and h (p -4 s) infer “ l (s -4 {q -4 r)).
Proof. Suppose H (p -4 (g -4 r)) and h (g -4 s). By AO h ((g -4 s) -4 ((p -4 (s -4
r)) —4 (p —> (<7-4 r)))). By MP h ((p -4 (s -4 r)) -4 (p -4 (g -4 r))). By R-D
H (p -4 (s —4 r)). Use similar reasoning for ii). ■
Proofs for the following two derived rules are easily constructed and will be omitted.
R-DR4 (Rejection by consequent strengthening (or equivalence), R-CS) From H (p -4q)
and h (r —4q) infer H (p —4r ); and from H (p —Hq -4r)) and
h (s -4r) infer “ l (p -4(q -4s)).
R-DR5 (Rejection by substitution of equivalents, R-SE) From h (p F4 q) and
H (...p...) infer H (...g...).
Aristotle’s Modal Syllogisms
255
R-DR6 (Rejection by implication introduction, R-II) From hp and infer
<?)•
Proof. Suppose h p and H q. By AO h (p —>■ ((p —>• qr) —>• q)). By MP h ((p —> q) -A q). By
R-D H (p-M). ■
Given the above apparatus we are able to show how the four syllogisms referred to at
Prior Analytics 26a2-9 are rejected in LA. This is Lukasiewicz’s translation from [1957,
p. 67].
If the first term belongs to all the middle [Aba], but the middle to none of the
last [Ecb], there will be no syllogism of the extremes; for nothing necessary
follows from the terms being so related; for it is possible that the first should
belong to all as well as to none of the last, so that neither a particular nor a
universal conclusion is necessary. But if there is no necessary consequence
by means of these premises, there cannot be a syllogism. Terms of belong to
all: animal, man, horse; to none: animal, man, stone.
The four syllogisms are (Aba —► (Ecb —► a;)), where x is lea, Oca, Aca or Eca.
We shall give rejection deductions to establish the rejection of the first two (AEI-1 and
AEO-1) and then use derived rule R-CS to show the last two (AEA-1 and AEE-1) are
rejected . 8
Theorem 1.6. (Rejection of AEI-1) H (A6o -4 ( Ecb -4 lea )).
Proof.
1. ''(Acb -4 (Aab -4 lac)) (by Rl)
2. h (Iac -4 (Acb —> (Aab -4 lac))) (by AO)
3. H /ac (from 1 and 2 by R-D)
4. h Acc (by A1 and US)
5. H (Acc -4 lac) (from 3 and 4 by R-II)
6 . H (Acb —4 lab) (from 5 by R-US)
7. -*(Eab -4 Ocb) (from 6 by R-RV and R-SE)
8 . ''(Acb -4 lac) (from 5 by R-US)
9. ''(Eac -4 Ocb) (from 8 by R-RV and R-SE)
10. H (Eab -4 (Eac —> Ocb)) (from 7 and 9 by R-S)
11. H (Acb —> (Eac -4 lab)) (from 10 by R-RV)
12. H (Aba —» (Ecb -4 lea)) (from 11 by R-US) ■
8 In the above passage Aristotle uses the semantic counterpart of this two-stage syntactic process. First,
he shows by his counterexample that {Aba, Ecb, Aca) and {Aba, Ecb, Eca} are semantically consistent,
from which it follows that neither of the particulars Oca and lea is a semantic consequence of {Aba, Ecb}.
Secondly, since the universal claims Eca and Aca are stronger than Oca and lea, respectively, they cannot
be a semantic consequence of {Aba, Ecb}. Aristotle is using what W. D. Ross [1949, p. 302] calls a “proof
by contrasted instances,” to show a pair of premises is, in Jonathan Lear’s [1980, p. 54] terms, “semantically
sterile”.
256
Fred Johnson
Theorem 1.7. (Rejection of AEO-1) ’’(Aba —KEcb —>Oca)).
Proof.
1. H {Acb (Aablac)) (ByRl)
2. ''((Acb —> ( Eac —> Oab)) (from 1 by R-RV and SE)
3. ''(Aba -4 (Ecb -» Oca)) (from 2 by R-US) ■
Theorem 1.8. (Rejection of AEA-1 and AEE-1) i) H (A6a —> (i?c6 —> Aca)); and ii)
-'(Aba —> (Ecb -4 Eca)).
Proof.
1. H (Aba —» (.Ecb —» 7ca)) (by theorem 1.6)
2. h (Aca —> lea) (by Sub-a, US)
3. ''(Aba — > (Ecb — > Aca)). (i, from 1 and 2 by R-CS)
4. H (Aba -> (Ecb -4 Oca)) (by theorem 1.7)
5. '-(Eca —> Oca) (by Sub-a, US)
6. -'(Aba(EcbEca)) (ii, from 4 and 5 by R-CS) ■
The following passage clearly shows that Ross favors Lukasiewicz’s method of reject¬
ing the AEx-ls over Aristotle’s. On p. 302 of [1949] Ross says:
... [Aristotle] gives no reason (my italics) for this [claim that no conclusion is
yielded by the premises of AEx-1], e.g. by pointing out that an undistributed
middle or an illicit process is involved; but he often points to an empirical
fact. ... instead of giving the reason why All B are A, No C is B yields
no conclusion, he simply points to one set of values for A, B, C (animal,
man, horse) for which, all B being A and no C being B, all C is in fact A,
and to another set of values (animal, man, stone) for which, all B being A
and no C being B, no C is in fact A. Since in the one case all C is A, a
negative conclusion cannot be valid; and since in the other case no C is A, an
affirmative conclusion cannot be valid. Therefore there is no valid conclusion
(with C as subject and A as predicate).
Aristotle is reasoning as follows. It is true that all men are animals, it is true that no
horses are men, and it is true that all horses are animals (and thus false that no horses are
animals and false that some horses are not animals). So neither Eca nor Oca is a logical
consequence of Aba together with Ecb. Since it is true that all men are animals, it is true
that no stones are men, and it is true that no stones are animals (and thus false that all
stones are animals and false that some stones are not animals), it follows that neither Aca
nor lea is a logical consequence of Aba together with Ecb.
Lukasiewicz also objects to Aristotle’s reasoning, claiming in [1957, p. 72] that it:
introduces into logic terms and propositions not germane to it. ‘Man’ and
‘animal’ are not logical terms, and the proposition ‘All men are animals’ is
not a logical thesis. Logic cannot depend on concrete terms and statements.
If we want to avoid this difficulty, we must reject some forms axiomatically.
Aristotle’s Modal Syllogisms
257
But Aristotle’s procedures have support among modern logicians. Robin Smith
[ 1989, p. 114] regards Aristotle’s reference to animals, men and horses as a reference to
a “countermodel” and says “countermodels are the paradigmatic means of proving inva¬
lidity for modern logicians.” In the surrounding text Smith refers to Jonathan Lear [1980,
pp. 54-61 and pp. 70-75] who defends Aristotle’s techniques against criticisms by
Lukasiewicz and Geach [1972], In the following sections we shall make extensive use
of formal countermodels to show the invalidity of apodeictic and contingent syllogisms.
Such models may also be used to show the invalidity of assertoric syllogisms.
The following passage from the Prior Analytics 27bI2-23, quoted and discussed by
Lukasiewicz on p. 70 of [1957], illustrates another method Aristotle uses to reject in¬
ferences. Ross [1949, p. 304] calls it an argument “from the ambiguity of a particular
proposition.” A better name for the reasoning is “rejection by premise weakening”. Ross
points out that this method of rejection is also used by Aristotle at 26b 14-20, 27b27-28,
28b28-31,29a6 and 35b 11.
Let M belong to no N, and not to some X. It is possible then for N to belong
either to all X or to no X. Terms of belonging to none: black, snow, animal.
Terms of belonging to all cannot be found, if M belongs to some X, and does
not belong to some X. For if N belonged to all X, and M to no N, then M
would belong to no AT; but it is assumed that it belongs to some X. In this
way, then, it is not possible to take terms, and the proof must start from the
indefinite nature of the particular premise. For since it is true that M does
not belong to some X, even if it belongs to no X , and since if it belongs to
no X a syllogism is not possible, clearly it will not be possible either.
Given the semantic consistency of {No snow is black, Some animals are not black.
No animal is snow } we know by half of the “contrasted instances” argument that neither
‘Some animal is snow’ nor ‘All animals are snow’ is a logical consequence of ’No snow
is black’ together with ’Some animals are not black.’ So, a “countermodel” is given for
the inferences from Enm and Oxm to Ixn or Axn. To show that neither Oxn nor Exn
is a semantic consequence of Enm and Oxm , Aristotle relies on two facts: i) neither
Oxn nor Exn is a semantic consequence of Enm and Exm\ and ii) Oxm is a semantic
consequence of Exm.
In LA a purely syntactic rejection of the “implicational syllogisms” (Enm -A ( Oxm —>
Oxn)) and ( Enm —> ( Oxm -4 Exn)) is given by using R-AW.
Theorem 1.9. (Rejection of EOO-2 and EOE-2) i) H (Enm -4 (Exm —> Oxn))', and ii)
H (£nm -4 (Oxm -4 Exn)).
Proof.
1. H (Aba -4 ( Ecb -A lea)) (by theorem 1.6)
2. Eca —* (Ecb -A Oba)) (from 1 by R-RV and R-SE)
3. h (Ecb -4 Obc) (by Con(pa) and US)
4. H ( Eca -4 ( Obc -4 Oba)) (from 2 and 3 by R-AW)
5. h (Eca -4 Eac) (by Con and US)
258
Fred Johnson
6 . ~*(Eac —4 ( Obc -4 Oba)) (from 4 and 5 by R-AW)
7. ~'(Enm -4 (Oxm -4 Oxn )) (i, from 4 by R-US)
8 . h (Exn -4 Oxn) (by Sub-a and US)
9. ''(Enm —4 (Oxm —4 Exn)) (ii, from 7 and 8 by R-CS) ■
Up to this point we have rejected elementary wffs of form (xi —4 (x 2 —4 ... (x„ —4
y) ■ ■ ■) where n <2. For Lukasiewicz’s system to be fully Aristotelian he must show how
elementary sentences, where n > 2, are rejected. We illustrate such a rejection.
Theorem 1.10. (Rejection of an AAAA mood) ''(Aab — 4 (Abc — 4 (Adc —4 Aad))).
Proof.
1. H (Acb —4 ( Aab — 4 lac)) (by Rl)
2. h (Acb —4 (Aba —4 /ac)) (by Bramantip)
3. ~'((Acb —4 (Aba —4 Iac))mc(Acb -4 (Aab —4 7ac)) (from 2 and 1 by R-II)
4. l ~((Aba -4 Aab) -4 (( Acb -4 (Aba -4 Iac))mc(Acb -> (Aab —4 lac))) (by AO)
5. H (Aba —4 Aab) (from 3 and 4 by R-D)
6 . h Aaa (by Al)
7. H (Aaa —4 (Aba -4 Aab)) (from 6 and 5 by R-II)
8 . H (Aaa -4 (Aaa -4 (Aba -4 Aab))) (from 7 and 5 by R-II)
9. H (Aaa -4 (Aaa —4 (Ada -4 Aad))) (from 8 by R-US)
10. H (Aaa -4 (Aac -4 (Adc -4 Aad))) (from 9 by R-US)
11. ''(Aab -4 (Abe -4 (Adc -4 Aad))) (from 10 by R-US) ■
Lukasiewicz’s system for the assertoric syllogistic has “100% Aristotelicity”, to use
McCall’s expression. This means that every 2-premised syllogism deemed valid by Aris¬
totle is asserted in Lukasiewicz’s system, and every 2-premised syllogism deemed invalid
by Aristotle is rejected in Lukasiewicz’s system. We shall see below that McCall’s L-X-M
calculus also has 100% Aristotelicity though his Q-L-X-M calculus does not.
2 LUKASIEWICZ’S MODAL SYSTEM, LM
Lukasiewicz developed his system for the assertoric syllogistic by using the non-modal
propositional calculus, what he calls the “theory of deduction,” as a “base logic”. Fol¬
lowing the procedure used in Hughes and Cresswell’s [1968] and [1996], we simplified
Lukasiewicz’s presentation of his system by simply using axiom A0 to provide his “ba¬
sis”. Lukasiewicz’s approach to Aristotle’s modal logic is to develop a modal proposi¬
tional logic (with quantifiers), which we refer to as the “LM system”, that will enable him
to present Aristotle’s work on the modal syllogisms . 2 * * * * * * 9
The following sentences are tautologies in LM, modifying Lukasiewicz’s notation in a
natural way: 1) ((p — 4 q) — 4 (Mp -4 Mq)) and 2) ((p — 4 q) -4 (Lp —4 Lq)), reading M
and L as ‘it is possible that’ and ‘it is necessary that’, respectively. The following passages
on p. 138 of [Lukasiewicz, 1957] attempt to show that the “M-law of extensionality” (1)
and the “L-law of extensionality” (2) are endorsed by Aristotle.
9 See [1961] for Smiley’s extensions of Lukasiewicz’s work on LM.
Aristotle’s Modal Syllogisms
259
First it has to be said that if (if a is, /3 must be), then (if a is possible, (3 must
be possible too). [34a5-7]
If one should denote the premises by a, and the conclusion by /?, it would
not only result that if a is necessary, then /3 is necessary, but also that if a is
possible, then is possible. [34a22-24]
It has been proved that if (if a is, /3 is), then (if a is possible, then f} is
possible). [34a29-31]
A more natural reading of these passages is that they show that Aristotle endorsed both
3) (L(p -4 g) — 4 (Mp — 4 Mq )) and 4) ( L(p -A q) — 4 ( Lp —4 Lq))3°
That 1) - 4) are tautologies in LM is seen by considering the following four-valued
truth tables.
Table 2. Four-valued truth tables for —4, L and M
-4
1
2
3
4
—1
M
L
*1
1
2
3
4
4
1
2
2
1
1
3
3
3
1
2
3
1
2
1
2
2
3
4
4
1
1
1
1
1
3
4
Among the four truth values 1 to 4, 1 is the only designated value, marked with an
asterisk in its entry in the first column on the table. A sentence x in the LM-system is a
tautology iff for every input of values the output value is always the designated value 1 .
Theorem 2.1. (L-law of extensionality) ((p —4 q) —4 (Lp -4 Lq )) is a tautology.
Proof. Suppose ((p —4 g) —4 (Lp -4 Lq)) is assigned a value other than 1. Then i)
(p -4 q) is not assigned 4 and ii) (Lp -4 Lq) is not assigned 1, and iii) the value assigned
to (p -4 q) is not the value assigned to (Lp —4 Lq). By i) p is not assigned 1 and q is not
assigned 4. By ii) Lp is not assigned 4 and thus p is assigned neither 3 nor 4. And by ii)
Lp is not assigned the same value as Lq. So p is assigned the value 2 and q is assigned
the value 3. Then (p —4 q) and (Lp -4 Lq) are assigned the same value, which conflicts
with iii). ■
Proofs that 1), 3) and 4) are tautologies are not required for our purposes, and we omit
the straightforward proofs.
McCall [1963, pp. 31-32] points out that Lukasiewicz’s use of the L-law of exten¬
sionality yields highly unAristotelian results. For example, using McCall’s notation,
Camestres LXL (‘Necessarily all c are 6 ; no a are 6 ; so (necessarily) necessarily no a
10 See [Hughes and Cresswell, 1968, pp. 29-30] for a discussion of this sentence, an axiom in Robert Feys's
System T.
260
Fred Johnson
are c’), Baroco LXL ('Necessarily all c are b\ some a are not 6 ; so necessarily some a
are not c’), Barbara XLL (‘All b are c; necessarily a are b; so necessarily all a are c’) and
Ferio XLL (‘No b are c; necessarily some a are b ; so necessarily some b are c’), when con¬
strued as “implicational syllogisms”, are asserted in Lukasiewicz’s L-system even though
Aristotle rejects all of them.
Following McCall we use ‘XXX’ after the name of a syllogism to indicate that the
syllogism is a plain, assertoric syllogism. So, for example, Camestres XXX has form
‘All c are b ; no a are 6 ; so no a are c’. Camestres XXX, Baroco XXX, Barbara XXX
and Ferio XXX are asserted in Lukasiewicz’s assertoric system. So, given the following
theorem, Camestres LXL, Baroco LXL, Barbara XLL and Ferio XLL are asserted in
Lukasiewicz’s L-system.
Theorem 2.2. i) h ((p —» (q —> r)) —> (p —> (Lq —> Lr)))', and ii) h ((p -> (g —> r)) —>
(Lp ->(?-► Lr))).
Proof.
1. h ((q —» r) —> (Lq —> Lr)) (by theorem 2.1)
2. k (((<7 -> r) —» (Lg Lr)) —> ((p -> (g —> r)) —> (p —>■ (Lg —>■ Lr))) (by AO)
3. h ((p -> (g -4 r)) -> (jp -» (Lg -> Lr))) (i, from 1 and 2 by MP)
4- h (((P (9 0) (P (-£-9 i' 7 ')) -> ((9 -»(?->• O) -> (Lq -> (p ->
Lr))) (by AO)
5. h ((g —> (p —f r)) -> (Lq —► (p -> Lr))) (from 3 and 4 by MP)
6 . h ((p —>(q ->r)) ->(Lp ->(q —>Lr))) (ii, from 5 by US) ■
One of the virtues of McCall’s L-X-M calculus, discussed below, is that Camestres
LXL, Baroco LXL, Barbara XLL and Ferio XLL are rejected in it. But before we examine
McCall’s system we look briefly at some recent systems of modal predicate logic that have
been used to attempt to understand Aristotle’s work on the modal syllogisms.
3 MODERN MODAL PREDICATE LOGIC
It is natural to try to view Aristotle’s modal logic through the eyes of modern modal
monadic first order predicate logic." On pp. 18-22 McCall refers to Albrecht Becker’s
[1933] 12 and works by others who have tried to do this. On pp. 176-181 Patterson dis¬
cusses Ulrich Nortmann’s [ 1990] attempt to do this. Patterson points out that the Kripkean
“possible worlds semantics” used by Nortmann does not conform with Aristotle’s onto¬
logical principles. I agree. McCall argues that all uniform readings of Aristotle’s modal
propositions as sentences in a modal first order predicate logic will make some valid Aris¬
totelian syllogisms invalid or will make some invalid Aristotelian syllogisms valid. I also
agree with McCall and will give some examples that support his position.
11 For recent books that contain sections on modal predicate logic see [Hughes and Cresswell. 1996], [Fitting
and Mendelsohn, 1998], [Girle, 2000] and [Bell etai, 2001],
12 See [Bochenski, 1963, pp. 57-62] for a useful discussion of Becker’s work.
Aristotle’s Modal Syllogisms
261
To illustrate how invalid Aristotelian inferences may be made valid consider
Bocardo LXL, (that is, ‘ LObc,Aba\ so LOac’, using McCall’s notation). Suppose we
translate it into modal predicate logic as: ‘3x(Bx f\O^Cx)-, Vx(Bx —> Ax); so 3x(AxA
□->Cx)’ (that is, ‘There is an x such that x is a B and x is necessarily not a C; for all x
if x is a B then x is an A\ so there is an x such that x is an A and x is necessarily not
a C’). We are using one of Becker’s two methods for translating LO sentences. Using
“singular sentences” such as Bm (read as m is a B, for ‘Max is a bear’, for example)
and familiar rules such as Existential Instantiation (El) 13 , Universal Instantiation (UI) and
Existential Generalization (EG) together with propositional calculus (PC) inferences we
may construct a deduction for Bocardo LXL, which Aristotle considered to be invalid. 14
Proof.
1. 3x(Bx A D-iCx) (premise)
2. Vx(Bx — > Ax) (premise)
3. (Bm A □ ->Cx) (from 1 by El)
4. (Bm —> Am) (from 2 by UI)
5. (Am A □ ~>Cx) (from 3 and 4 by PC)
6. 3x(Ax A □-'Cx) (from 5 by EG) ■
To illustrate how valid Aristotelian inferences may be made invalid, consider Bo¬
cardo LLL, (that is, ‘LObc; LAba; so LOac’, using McCall’s notation). Using another
Becker translation of LO sentences and a Becker translation of LA sentences the argu¬
ment amounts to this: ‘Vx(Cx -> OBx);3x(OAx A OsBx); so 3x(DAx A CHCx),
call it the “the MPredC argument”. Aristotle at [30a6-14] gives a proof by ecthesis to
show that Bocardo LLL is valid. But using the semantics for the modal system, S5, the
translated argument is S5-invalid. For suppose there are only two possible worlds w\ and
ui 2 , where each world “sees” each world (including itself). If “the MPredC argument”
is S5-valid then the following modal propositional calculus argument is S5-valid, call it
the “the MPropC argument”: ‘((Cm —> DBm) A (Cn —> OBn))- ((OAm A □-> Bm) V
(OAn A O-iBn)); so ((Oim A D-iCm) V (OAn A D-iCn))’. But then a countermodel
is constructed by: i) letting Am, Bn and Cn be true in world ui\ ; ii) letting Bm, Cm and
An be false in w \; iii) letting Am, Cm and Bn be true in world W 2 ; and iv) letting Bm,
An and Cn be false in world W 2 - Then in w i (Cm —» DBm) is true, (Cn -A DBn) is
true, (□ Am A □-> Bm) is true, OAm A O-iCm) is false, and (OAn A O-iCn) is false.
So “the MPropC argument” is S5-invalid. So “the MPredC argument” is invalid.
The same countermodel may be used to invalidate the argument that results by replac¬
ing the premise Vx(Cx —> OBx) in “the MPredC” argument with Vx(Cx —> OBx).
Geach [1964, p. 202] makes the following remarks about McCall’s comments list of
seven “Becker-type interpretations”:
b In [Johnson, 1993] Aristotle’s proofs by ecthesis are treated as essentially proofs by Existential Instantia¬
tion, For alternative accounts of proofs by ecthesis see [Thom, 1993] and [Smith. 1982],
14 Paul Thom in [1991] argues that Aristotle made a mistake in regarding Bocardo LXL as valid. Thom
contrasts his views with those in [Johnson, 1989], [Patterson, 1989], [Patterson. 1990] and [van Rijen, 1989],
262
Fred Johnson
Here McCall has not proved what he claims: namely that no Becker-type in¬
terpretation will secure simultaneously the validity of Barbara LLL and LXL,
the invalidity of Barbara XLL, and the simple conversion of LI propositions
(C Llab LIba). For all of these results are obtained if we combine reading (i)
of LA from McCall’s list with reading (iii) or equivalently (iv) of LI.
McCall’s list on p. 21 of Becker type interpretations is given on table 3.
Table 3. Seven Becker-type interpretations
Universal
Particular
(i)
Vx(Ax —¥ OBx)
3 x(Ax A OBx)
(ii)
0\/x{Ax —> Bx)
□3 x{Ax A Bx)
(iii)
VxD(Ax —> Bx)
3 xO(Ax A Bx)
(iv)
Vx(D Ax -> OBx)
3x(pAx A OBx)
(v)
Vx(0 Ax —> OBx)
3x(OAx A OBx)
(vi)
Vx(0 Ax —> Bx)
3 x{OAx A OBx)
(vii)
Vx(DAx -4 Bx)
3x(DAx A Bx)
McCall finds interpretations (i) and (ii) in [Becker-Freyseng, 1933], (ii) in [von Wright,
1951], (i) to (v) in [Sugihara, 1957a] and [Sugihara, 1957b], and all but (v) in [Rescher,
1963],
This is what McCall says about these seven interpretations:
None of these interpretations does justice to Aristotle’s system. Not one of
them even simultaneously provides for the validity of Barbaras LLL, the
invalidity of Barbara XLL, and the convertibility of the particular premise
‘Some A is necessarily B' into ‘Some B is necessarily A’.
And McCall is correct. Geach is in effect proposing two more interpretations in ad¬
dition to the seven on the list. Let us call one of them (viii), where LAab is translated
as \/x(Ax -> OBx) and Llab is translated as 3xO(Ax A Bx ). As Geach says, the
other one is essentially the same as it. But interpretation (viii) produces results that
are not Aristotelian. For example, if Darii-LXL, valid for Aristotle, is translated using
interpretation (viii) the resulting argument is S5-invalid. McCall is looking for an in¬
terpretation that provides “100% Aristotelicity”. Geach (p. 202) invites the reader to
consider an interpretation of McCall’s LAab and LOab as sentences of an extended as-
sertoric syllogistic, call it the “G-system", that allows sentences to be formed by using
complex terms, terms of form A p (necessarily p) and pp (possibly p), where p is a simple
term. McCall’s LAab, LEab, Llab and LOab are translated into the G-system as AaXb,
Eapb, IXaXb and Oapb respectively. Geach (p. 202) says:
A decision procedure for this calculus can easily be devised: write every
formula so that A-terms and //-terms appear instead of categoricals prefaced
Aristotle’s Modal Syllogisms
263
Figure 1. The invalidity of Darii LXL in the G-system
Ac a
with L, add an antecedent of the form CAXaa [that is, (AXaa —>] for each
A-term and one of the form CAafia [that is, (Aafia ->] for each /i-term,
and apply Lukasiewicz’s decision procedure for the plain syllogistic to the
resulting formula.
So, for example, to determine whether Bocardo LXL (that is, ‘LObc; Aba; so LOac’)
is syntactically accepted or syntactically rejected we form the following sentence in the
G-system: (Acfic —> (Obfic —» (Aba —s- Oafic))). Following
Lukasiewicz’s decision procedure on pp. 121-126 of 11957], we form an elementary
sentence consisting of affirmative simple sentences that is deductively equivalent to it:
{Ac/ic —> {Aa/ic —> (,46a —» Abfic))) or (by interchanging terms) ( Ab/ib —» (Acfib —>
{Aac -> Aa/ib))). The latter sentence fits subcase (d) of the fifth case (p. 124):
The consequent is Aab, and there are antecedents of the type Aaf with /
different from a. If there is a chain leading from a to b the expression is
asserted on the ground of axiom 3 [our A3, above], the mood Barbara; if
there is no such chain, the expression is rejected.
Since a is linked to b by the chain {Aac, Acfib}, {Abfib —T (Acfib —> {Aac —>
Aaub ))) is accepted. So Bocardo LXL is accepted in the G-system. But for Aristotle
Bocardo LXL is valid.
Since questions of validity in the G-system are reduced to questions of validity in the
assertoric syllogistic, the familiar Euler diagrams provide a technique for determining
whether or not arguments are valid. So, for example, the diagram in figure 1 displays the
invalidity of Darii LXL, {LAbc —» {lab -» Llac )). Since circle b is included in circle Ac,
LAbc is true. Since circle a overlaps circle 6, lab is true. Since circle A a does not overlap
Ac, Llac is false. When constructing such diagrams these conditions must be met: for
every term x, the Ax circle is included in or equal to the x circle, which is included in or
equal to the fix circle. These conditions are natural since whatever is necessarily x is x,
and whatever is x is possibly x.
The diagram in figure 2 displays the invalidity of Cesare LLL, {LEcb -A
{LAab —» LEac )). LEcb is true since circle c does not overlap circle fj,b\ LAab is
true since circle a is included in circle A b, which is identical to circle fib; and LEac is
false since circle a overlaps circle fic.
264
Fred Johnson
Figure 2. The invalidity of Cesare LLL in the G-system
fie fib , Ab
Geach does not claim that his G-system has “100 percent Aristotelicity”. He says on p.
203 of [1964] that it “can fit in most of Aristotle’s results about syllogisms de necessario
But table 4 shows that the G-system does not get high marks. “V” occurs in a cell if and
only if the relevant syllogism is valid for Aristotle, and “Gc” occurs in a cell if and only
if the G-system’s judgment about the acceptance or rejection of the relevant syllogism
is in conflict with Aristotle’s. So, for example, the “Gc” in the Darii/LXL cell means
that Darii LXL is rejected in the G-system though Aristotle accepts it. The “Gc” in the
Bocardo/LXL cell means that Bocardo LXL is accepted in the G-system though Aristotle
rejects it. The G-system’s Aristotelicity is ((3 x 14) - 13) 4 (3x 14) or about 69%.
Table 4. Aristotle’s system vs. the G-system
LLL
LXL
XLL
Figure 1
Barbara
V
V
Celarent
V
V
Darii
V
V,Gc
Ferio
V
V
Figure 2
Cesare
V,Gc
V,Gc
Camestres
V,Gc
V,Gc
Festino
V,Gc
V,Gc
Baroco
V,Gc
Figure 3
Darapti
V
V,Gc
V,Gc
Felapton
V
V
Disamis
V
V,Gc
Datisi
V
V,Gc
Bocardo
V
Gc
Ferison
V
V
Geach’s G-system and Lukasiewicz’s LM illustrate two approaches to understanding
Aristotle’s work on modal logic. Martha Kneale on p. 91 of [Kneale and Kneale, 1962]
poses a dilemma for students of Aristotle given her belief that there are only two ap¬
proaches to Aristotle’s work.
Aristotle’s Modal Syllogisms
265
If modal words modify predicates [Geach’s de re approach is taken], there is
no need for a special theory of modal syllogisms. For these are only ordinary
assertoric syllogisms of which the premises have peculiar predicates. On the
other hand, if modal words modify the whole statements to which they are
attached [Lukasiewicz’s de dicto approach is taken], there is no need for a
special modal syllogistic since the rules determining the logical relations be¬
tween modal statements are independent of the character of the propositions
governed by the modal words.
McCall agrees with Kneale that the two approaches described above are inadequate.
And he devises a third approach that is designed to “catch the fine distinctions Aristotle
makes between valid and invalid syllogisms (p. 96)”.
4 MC CALL’S L-X-M SYSTEM
The basis for L-X-M includes that of LA together with the following primitive symbols,
formation rules, definitions, axioms and transformation rules. Only some of the rejection
axioms are given here. The partial list is big enough to illustrate how rejection deductions
are constructed in L-X-M. For the full list of rejection axioms see [McCall, 1963] or
[Johnson, 1989],
Primitive symbols
monadic operator L
Formation rules
FR1' If Q u is a quantifier and x and y are term variables then Q u xy
expression .
FR2' If p is a categorical expression then ->p is a categorical expression
FR3' Categorical expressions are wffs .
FR4' If p and q are wffs then ->p and ( p —> q) are wffs .
FR5' The only wffs are those in virtue of FR1' to FR4'.
So, for example, Aab is a categorical expression by FR1', so -> Aab is a categorical
expression by FR2', so Aab is a categorical expression by FR2', so L->-iAab is a wff
by FR2', so -iL-<-<Aab is a wff by FR4'. Note that LLAab is not a wff.
Definitions
DefM Mp=df~^L~^p
Assertion axioms
Use AO, Al, A3 and A4 from system LA. Change A2 for LA from h Iaa to h LIaa. Then
add the following axioms.
is a categorical
and Lp is a wff.
266
Fred Johnson
A5 (Barbara LXL)
A6 (Cesare LXL)
A7 (Darii LXL)
A8 (Ferio LXL)
A9 (Baroco LLL)
A10 (Bocardo LLL)
All (LIconversion)
A12 (LA subordination)
A13 (LI subordination)
A14 (LO subordination)
h (LAbc — ► (Aab -4 LAac))
l ~(LEcb —► (Aab —> LEac ))
h (LAbc -4 (lab —> Llac))
h (LEbc -4 (lab -4 LOac))
h (LAcb -4 (LOab -4 LOac))
h (LObc -4 (LAba -4 LOac))
h (L/a6 -4 LIba)
h (LAab — > Aaft)
h (LIab -4 7a6)
h (LOab -4 Oa6)
Assertion transformation rules
Use the assertion transformation rules ArI to Ar 3 from LA and add the following rule.
Ar 4 (Assertions involving doubly negated categorical expressions, DN) From h (... p...)
infer h (... -<->p...) and vice versa, if p is a categorical expression. (So, for exam¬
ple, from h (LAab -4 LAab) infer h (LAab -4 L^Aab) by DN. By using SE we
may infer that h (LAab -4 ->-^LAab) given h (LAab -4 LAab).)
Rejection axioms
Use R1 from system LA and add the following rejection axioms.
R2 (*5.21, p. 58) H (LAbb -4(MAab ->(Aac -4(LAca -4(LAbc -4LAac)))))
R3 (*5.3, p. 58) H (LAaa -4(LAcc ->(MAac -J(LAca 4-Aac)))) 15
R4 (*5.6, p. 64) H (LAaa —KLAbb ->(LAcc -4(LAab -4(MAba -4
(MAbc -4(LAcb -4lac)))))))
Page references are to McCall’s [1963]. McCall uses asterisks to refer to rejections.
Rejection transformation rules
Use rejection transformation rules Rr1-Rr 4 as well as the following rule.
Rr 5 (Rejections involving doubly negated categorical expressions, R-DN) From
H (.. .p ...) infer “ l (... -i-ip...) and vice versa, if p is a categorical expression. (So,
for example, from H L->-La6 infer H L/a&.)
We imitate the discussion of Lukasiewicz’s LA system by proving various “immediate
inferences”. Oppositions, conversions, subalternations and subordinations are listed.
Theorem 4.1. (Apodeictic oppositions, Ap-opp) i) h (^LAab 44 MOab)\
ii) h (^MOab 44 LAab)\ iii) h (-^LEab 44 Mlab ); iv) l ”(-i Mlab 44 LEab)\ v)
15 Correction: on p. 273 of [1989] change 4 LAac in *5.3 to 4 Aac.
Aristotle’s Modal Syllogisms
267
h (->L/a6 MEab); vi) h (-i MEab o Llab ); vii) h (-^LOab O MAab); and viii)
h (^MAab -o- LOab).
Proof.
1. h (^LAab <-> -i LAab ) (by AO)
2. h (->LAab -iL->-iAa6) (from 1 by DN)
3. h (-iLAafr -B- MOab) (i, from 2 by DDI, given Def M and Def 0)
4. h (-i MOab -o- ->-^LAab) (from 3 by RV)
5. h (-r-iLAa6 LAab) (by AO)
6. h (->MOafr f* LAab) (ii, from 4 and 5 by SE)
7. h (^LEab o -> LEab ) (by AO)
8. h (-iLEab «-» -^L^^Eab) (from 7 by DN)
9. h (-iL£a6 (from 8 by DDI, given Def E)
10. h (-*LEab <-> -i L^Iab) (from 9 by DN)
11. h (-iLEab Mlab) (iii, from 10 by DDI, given Def M)
12. h (-iAf/a6 -i-iLEab) (from 11 by RV)
13. h (-i-iLjBa6 +-» LEab) (by AO)
14. l "(-i Mlab LEab) (iv, from 12 and 13 by SE) ■
Use similar reasoning for the other four asserted biconditionals.
Theorem 4.2. (Apodeictic conversions, Ap-con) i) h (LEab —» LEba);
ii) h (LIab —> LIba ); iii) h (MEab —» MEba); and iv) h (MIab —> MIba).
Proof.
1. h (Llab —> LIba ) (ii, by All)
2. h (-i LIba —> -i Llab) (from 1 by RV)
3. h (MEba —> MEab) (from 2 by SE, Ap-opp)
4. h (MEab —> MEba) (iii, from 3 by US)
5. *~(LEcb —> (Aaft —> LEac)) (A6)
6. h Aaa(Al)
7. h (LEca —> LEac) (from 5 and 6 by AI, US, MP)
8. h (LEab —> LEba) (i, from 7 by US)
9. h (MIab —> MIba) (iv, from 8 by RV, SE, US) ■
Theorem 4.3. (Apodeictic subalternations, Ap-sub-a) i) h (LAab —> Llab)-,
ii) h (LEab -» LOab)-, iii) h (MAab Mlab)-, and iv) H (MEab ->MOab).
Proof.
1. h (LAab —r LIba) (by All)
2. h (LIba —> Llab) (by Ap-con, US)
3. h (LAab —>■ Llab) (i, from 1 and 2 by CW)
268
Fred Johnson
4. h (MEab —> MOab) (iv, from 3 by RV, SE)
5. h (LEbc —F (/ab —> LOac)) (by A8)
6. h (LEac —> (/aa -4 LOac)) (from 5 by US)
7. h ( LEab —> LOab) (ii, from 6 by AI, MP, US)
8. h ( MAab -4 Mlab) (iii, from 7 by RV, SE) ■
Theorem 4.4. (Apodeictic conversions per accidens , Ap-con(pa)) i) h (LAab -4
LIba)\ ii) h (MAab -4 MIba)\ iii) h {LEab -4 LOba); and iv) h (MEab -4 MOba).
Proof.
1. h (LAab -4 Llab) (by Ap-sub-a)
2. h (LIab -4 LIba) (by Ap-con)
3. h (LAab -4 ZJba) (i, from 1 and 2 by CW)
4. h (LEab -4 LEba) (by Ap-con)
5. h (LEba -4 LOba) (by Ap-sub-a, US)
6. h (LEab -4 LOba) (iii, from 4 and 5 by CW)
7. h (MAab -4 M/6a) (ii, from 6 by RV, SE, US)
8. h (M£a6 -4 MOba) (iv, from 3 by RV, SE, US) ■
Theorem 4.5. (Subordinations, Sub-o) i) h (LAab -4 Aab ); ii) h (Aab -4 MAab)\ iii)
h (LEab -4 .Ea6); iv) h (Eab -4 MEab)\ v) h (LIab -4 /a&); vi) H (/a6 -4 MIab)\ vii)
h (LOab -4 Oa&); and viii) h (Oab -4 MOab).
Proof.
1. h (LOab —> Oab) (vii, by A14)
2. h (Aab —> MAab) (ii, from 1 by RV, SE)
3. h (Aoa -4 MAaa) (from 2 by US)
4. Maa(byAl)
5. h MAaa (from 3 and 4 by MP)
6. h (LEbc —> (/ab —» LOac)) (by A8)
7. H ( MAac -4 (/ab -4 MIbc)) (from 6 by RV, SE)
8. h (MAaa -4 ( lab —> MIba)) (from 7 by US)
9. h (/ab -4 MIba) (from 5 and 8 by MP)
10. h (M/bo -4 Mlab) (by Ap-con, US)
11. h (lab —» Mlab) (vi, from 9 and 10 by CW)
12. h (LEab -4 £ab) (iii, from 11 by RV, SE, US)
Proofs of the other four subordinations are straightforward and are omitted.
Aristotle's Modal Syllogisms
269
We show that all of the entries marked with ‘V’ on table 4 and all of the entries marked
with a blank on table 5 correspond to asserted wffs in L-X-M. Proofs are streamlined by
assuming immediate inferences established above and any immediate inferences obtain¬
able from them by US. So, for example, in the proof of Barbari LXL from Barbara LXL
by CW in theorem LXL the subalternation h (LAac — 4 Llac ) is assumed.
Theorem 4.6. All unmarked LXL and XLL cells on table 5 represent asserted wffs.
Proof.
1. h (LAbc —4 ( Aab —4 LAac)) (Barbara LXL, by A5)
2. h (LAbc -4 ( Aab —4 Llac)) (Barbari LXL, from 1 by CW)
3. h (Acb —4 ( LAba —4 Llac)) (Bramantip XLL, from 2 by AI, CW, US)
4. h (LAbc —4 ( lab —4 Llac)) (Darii LXL, by A7)
5. *~(LAbc -4 ( Iba —4 Llac)) (Datisi LXL, from 4 by AS)
6. h (LAbc -4 ( Aba —4 Llac)) (Darapti LXL, from 5 by AS)
7. h (/ftc —4 (LAab —4 Llac)) (Disamis XLL, from 5 by AI, CW, US)
8. h (/cft -4 ( LAba —4 Llac)) (Dimaris XLL, from 4 by AI, CW, US)
9. h (Acb —4 (LAba -4 Llac)) (Darapti XLL, from 7 by AS)
10. h (L£ftc -4 (/aft -4 LOac)) (FerioLXL, by A8)
11. h (LEcb -4 ( lab -4 LOac)) (Festino LXL, from 9 by AS)
12. h (LEbc -4 ( Iba -4 LOac)) (Ferison LXL, from 9 by AS)
13. h (LEbc —4 ( Aba -4 LOac)) (Felapton LXL, from 11 by AS)
14. h (LEcb —4 ( Iba —4 LOac)) (Fresison LXL, from 9 by AS)
15. h (LEcb —4 ( Aba -4 LOac)) (Fesapo LXL, from 13 by AS)
16. h (LEcb -4 ( Aab —4 LEac)) (Cesare LXL, by A6)
17. h (LEbc —4 ( Aab —4 LEac)) (Celarent LXL, from 15 by AS)
18. h (Acb -4 (LEab —4 LEac)) (Camestres XLL, from 16 by AI, CW, US)
19. h (Acb -4 ( LEba —4 LEac)) (Camenes XLL, from 17 by AS)
20. h (LEbc —4 (Aab —4 LEac)) (Celaront LXL, from 16 by CW)
21. h (LEcb -4 (Aab -4 LOac)) (Cesaro LXL, from 15byCW)
22. h (Acb —4 (LEab -4 LEac)) (Camestrop XLL, from 18 by CW)
23. h (Acb —4 ( LEba —4 LEac)) (Camenop XLL, from 19 by CW) ■
Theorem 4.7. All unmarked LLL cells on table 5 represent asserted wffs.
Proof. Use A9, A10 and AS with theorem 4.6. So, for example, Barbara LLL is asserted,
since Barbara LXL is asserted and h (LAab —4 Aab). Disamis LLL is assserted, since
Disamis XLL is asserted and h (LIab —4 lab). ■
Theorem 4.8. All unmarked MXM, XMM, LMX and MLX cells on table 5 represent
asserted wffs.
270
Fred Johnson
Proof. Use theorem 4.6 and RV. So, for example, the assertion of Darii MXM is generated
from the assertion of Ferison LXL as follows. h (MAbc —► (lab —> Mlac )) since
h (LEac —» (lab -4 LObc)) (by RV and SE), since h (LEbc -4 ( Iba —» LOac )) (by
US). The assertion of Festino LMX is generated from the assertion of Celarent LXL as
follows. h (LEcb -4 ( Mlab -4 Oac)) since h (LEcb -4 (Aac -4 LEab)) (by RV and
SE), since h (LEbc -4 (Aa6 -4 LEacj) (by US). The assertion of Camenes MLX is
generated from the assertion of Fresison LXL as follows. h (MAcb —> (LEba -4 Eac)),
since h (Iac -4 ( LEba -4 LOcb)) (by RV and SE), since h ( LEba —» (lac -4 LOcb))
(by AI), since (LEcb —> (Iba —> LOac)) (by US). ■
4.1 Rejections in L-X-M
To reject the syllogisms not marked with a “V” on table 4, as well as other invalid in¬
ferences, McCall adds twelve rejection axioms to the list of rejection axioms for the LA-
system. We shall illustrate how some of these rejection axioms are used to reject some
wffs.
Theorem 4.9. (Rejection of Barbara XLL) H (Abc ->(LAab -4LAac)).
Proof. Recall that R2 = H a , where a = (LAbb -4 (MAab -4 (Aac -4 (LAca -4
(LAbc -4 LAac))))).
1. H (j (by R2)
2. h ((Aac -4 (LAca -4 LAac)) -4 a) (by AO)
3. H (Aac -4 (LAca -4 LAac)) (from 1 and 2 by R-D)
4. h (Aac -4 (LAca -4 LAaa)) (by A5 and US)
5. ^((Aac —^ (LAca — t LAaa)) —^ ((Aac — y (LAaa —y LAac)) —^ (Aac — y
(LAca —> LAac)))) (by AO)
6. h ((Aac -4 (LAaa -4 LAac)) —> (Aoc —7 (LAca -4 LAac))) (from 4 and 5 by
MP)
7. H (Aac -4 (LAaa -4 LAac)) (from 3 and 6 by R-D)
8. H (A6c -4 (LAab -4 LAac)) (from 7 by R-US) ■
Theorem 4.10. Baroco XMM and Bocardo MLX are rejected.
Proof.
1. H ( Abe —> (LAab -4 LAac)) (by theorem 4.9)
2. ~*(Abc —> (MOac -4 MOab)) (from 1 by theorem R-RV and R-SE)
3. H (Ac6 -4 (MOab -4- MOac)) (Baroco XMM, from 2 by R-US)
4. -'(MOac (LAab -4 06c)) (from 1 by R-RV and R-SE)
5. H ( MObc -4 (LAba -4 Oac)) (Bocardo MLX, from 4 by R-US) ■
Theorem 4.11. Barbara LMX, Baroco LXL and Bocardo XMM are rejected.
Proof. Recall that R3 = where a is (LAaa -4 (LAcc -4 (MAac —> (LAca -4
Aac)))).
Aristotle’s Modal Syllogisms
271
1. ~V (by R3)
2. h ((LAcc -A ( MAac —4 Aac )) —4 cr) (by A5)
3. H (LAcc —4 ( MAac —4 Aac)) (from 1 and 2 by R-D)
4. ^(LA&c -4 (MAab -4 Aac)) (Barbara LMX, from 3 by R-US)
5. H ( LAbc (Oac —4 LOab)) (from 4 by R-RV and R-SE)
6. H (LAcb —> (Oa6 -4 LOac )) (BarocoLXL, from 5 by R-US)
7. H (Oac -4 (MAafc -4 M06c)) (from 4 by R-RV)
8. -'{Obc -4 ( MAba -4 MOac)) (Bocardo XMM, from 7 by R-US) ■
Theorem 4A2. Barbari MLX, Bramantip LMX, Felapton XLL and Baroco XLL are
rejected.
Proof. Recall that R4 = H cr, where a = (LAaa -4(LAbb -4(LAcc -4LAab -4(MAba
-4(MAbc -4(LAcb -Aac)))))).
1. H a (by R4)
2. h ((MA6c -4 ( LAab -4 lac)) -4 a) (by AO)
3. "*( MAbc —4 ( LAab -4 lac)) (Barbari MLX, from 1 and 2 by R-US)
4. ^(LAab —4 ( MAbc —4 lac)) (from 3 by R-AI)
5. h (Ica -4 lac) (by Con)
6. ■•( LAab -4 ( MAbc -4 lea)) (from 4 and 5 by R-CS)
7. H (LAc6 —4 ( MAba -4 7ac)) (Bramantip LMX, from 6 by R-US)
8. H (£ac -4 ( LAab -4 L06c)) (from 3 by R-RV and SE)
9. H (E6c -4 (LAba -4 LOac)) (Felapton XLL, from 8 by R-US)
10. H (7J6c -4 Obc) (by Sub-a and US)
11. h (06c -4 (LAba -4 LOac)) (Baroco XLL, 9 and 10 by R-AW) ■
Our purpose in this section has been to illustrate how McCall’s rejection apparatus
works. In the next section we discuss this result: whatever is rejected by using McCall’s
rejection apparatus may be shown invalid by using countermodels. McCall’s [1963] con¬
tains no discussion of models.
5 SEMANTICS FOR L-X-M
In [Johnson, 1989] a semantics for McCall’s L-X-M is given. Validity is defined by using
models, asserted wffs in L-X-M are shown to be valid (that is, system L-X-M is sound
), and rejected sentences are shown to be invalid. So valid wffs in X-L-M are shown to
be accepted (that is, system L-X-M is complete ) since, as McCall shows, every wff in
L-X-M is either accepted or rejected. The presentation of the semantics here will benefit
from comments about it in Thom’s [1996] and Thomason’s [1993] and [1997]. 16
l6 For example, I borrow Thom’s use of “base conditions" and “superstructural conditions” to present what
he calls a “two-layered semantics”. And I borrow Thomason’s use of “Vjy ” to refer to a valuation relative to a
model.
272
Fred Johnson
The semantics for L-X-M extends the familiar semantics for the assertoric syllogistic
that assigns non-empty sets of objects to terms. To define the semantic notion of validity
we refer to models and valuations relative to models.
Definition 5.1. (model) M is a model iff M = ( W , n + ,q + , n~, q~), where W is a non¬
empty set and n + , q + , n~ , and q~ are functions that map terms into subsets of W and
satisfy the following “base conditions”, where + ( x) is short for n + ( x) U q + (x), and xoy
(x overlaps y ) is short for x ft y / 0:
B1 If / and g are any of the functions n + ,q + , q~ or n~ and f ^ g, then, for every term
x, f(x) fl g(x) = 0; and for every x, n + (x) U q + (x) U q~{x) U n~(x) — W
B2 For every x, n + (a;) ^ 0
B3 (For every x, y and z) if + (z) C n~(y) and + (x) C + (y) then + (x) C n~(z)
B4 If + (t/) C n + (z) and + (x) o + (y) thenn + (x) o n + (z)
B5 If + (y) C n~(z) and + (x) o + (y) then n + (x) o n~(z)
B6 If + (z) C n + (y) and n + (x) o n~(y) then n + (x) o n~(z)
For an intuitive grasp of the notion of a model think of W as the world, n + (a) as the set
of things in W that are essentially a, q + (a) as the set of things in W that are contingently
a and are a, n~ (x) as the set of things in W that are essentially non-a, and q~ (a) as the
set of things in W that are contingently not a and are not a.
Definition 5.2. (valuation) A valuation V is a function that assigns t or /, but not both,
to sentences, where: i) V (-ip) = t iff V (p) = /; and ii) V(p -> q) — t iff V(~'p) = t or
V{q) = /; and iii) V(L^p) = t iff V{Lp) = t.
Definition 5.3. (valuation relative to model M) Let Vm , a valuation relative to a model
M ,be a valuation that satisfies the following “superstructural conditions”:
51 (For every x and y) ViuiAxy) = t iff + (x) C + (y)
52 V M (Ixy) = t iff + (x) o +(y)
53 Vivt{LAxy) = t iff + (x) C n + (y)
54 VM(LIxy) = tiffn + (x) o n + (y)
55 Vp,i{L~>Axy) = t iff n + (x) o n~{y)
56 VM(L-*Ixy) = t iff + (x) C n~(y)
Definition 5.4. (valid) Let o be an L-X-M sentence, a is valid (|= a) iff, for every model
M, every valuation relative to M assigns t to o. a is invalid iff a is not valid.
In this section we shall construct models that show the invalidity of all of the syllo¬
gisms that correspond to marked cells on table 5. Exactly four models suffice to show
the invalidity of the invalid LXL and XLL models marked on these tables. Models con¬
structed by interchanging rows in these four models suffice to invalidate the remaining
invalid syllogisms mentioned on the table.
Table 5 agrees with table 7 on p. 43 of [McCall, 1963]. A cell on the former is marked
if and only if it is unmarked on the latter. The marks on McCall’s table indicate the
relevant syllogism is syntactically asserted in L-X-M. McCall’s discussion of L-X-M is
Aristotle's Modal Syllogisms
273
totally syntactic. He gives no formal semantics and thus no formal definition of validity.
But as shown in [1989], the syllogisms that are syntactically asserted in L-X-M are the
syllogisms that are valid in L-X-M and vice versa. The above theorems 4.6 and 4.7 pertain
to the unmarked cells on table 5.
Table 5.
Countermodels for L-X-M syllogisms
LLL LXL
XLL
MXM
XMM
LMX
MLX
Figure 1
Barbara
lac
4bc
4ab
3ab
2bc
Celarent
2ac
lba
2ab
Darii
lac
3ac
2bc
Ferio
2ac
2ab
lbc
Figure 2
Cesare
2ac
lba
2ab
Camestres
3ac
2ba
lab
Festino
2ac
2ab
lbc
Baroco
3ac
4ac
2ba
lab
4ba
Figure 3
Darapti
2cb
2bc
Felapton
2ac
3bc
lbc
Disarms
lea
2cb
2bc
Datisi
lac
2cb
2bc
Bocardo
4ac
2ac
3cb
4cb
lbc
Ferison
2ac
3cb
lbc
Figure 4
Bramantip
lea
2cb
3ba
Camenes
3ac
2ab
lab
Dimaris
lea
2cb
3ba
Fresison
2ac
3cb
lbc
Fesapo
2ac
3cb
lbc
Subalterns
Barbari
lac
3ab
2bc
Celaront
2ac
2ab
Cesaro
2ac
2ab
Camestrop
3ac
2ba
lab
Camenop
3ac
2ba
lab
We begin by constructing a model Ai i, presented by table 6, that shows the invalidity
of Barbara LXL. When giving such tables we use the following conventions: set brackets
are omitted when giving the range of a function, a blank cell indicates the range of the
relevant function is the empty set, for terms x other than those explicitly mentioned on
the table, n + (x ) = n + (a), q + {x) = q + {a), n~(:r) = n~(a) and q~{x) — q~{a), and
W — n + (a) U q + (a) U n~{a) U q~{a).
So, for example, given table 6 the set of things that are essentially a has only one
member, namely 1. The set of things that are c and are contingently c has two members:
1 and 2. The set of things that are essentially not b has no members. And the set of things
that are not d and are contingently not d has 3 as its only member. W — {1,2,3}.
Table 6 expresses a model. Base conditions B1 and B2, here and below, do not require a
comment. B3, B5andB6 are trivially satisfied since, for every x and y, + (x)C\n~(y) — 0.
274
Fred Johnson
Table 6. Model Mi
n +
q + n
Q
a
1
2
3
b
1 , 2,3
c
3
1,2
Suppose (y) C n + (z). Then 2 = 6. For all x, n + (x) on + (b). So B4 is satisfied.
Given model Mi', i) Fm,(j4Iic) = t since + (b) C + (c); ii) VMi(LAab) = t
since + (a) C n + (6); and iii) VmALAclc) = / since + (o) (Z n + (c). So Umj(A6c —>
(LAaB —» LAac)) = /. So ^ (Abe -4 ( LAab -4 LAac )). So Barbara XLL is invalid.
The invalidity of Barbara XLL is marked on table 5 by putting ‘lac’ in the Barabara/XLL
cell.
Aristotle’s informal counterexample for Barbara XLL at 30a28-30 uses terms ‘motion’,
‘animal’ and ‘man’. For Aristotle, Barbara XLL, construed as an inferential syllogism,
is invalid given the inference ‘All animals are (accidentally) in motion; all men are nec¬
essarily animal; so all men are necessarily in motion’. Aristotle takes the premises to be
true and the conclusion false, making Barbara XLL invalid.
By interchanging rows a and b in table 6 we may construct a model Mu, c expressed
by table 7 that shows that Ferio MLX invalid.
Table 7. Model Mu, c
q + n
9
a
1 , 2,3
b
1
2
3
c
3
1,2
In general, if a table satisfies conditions B1 to B6 so will a table that results from
the interchanging of its rows. For none of these conditions requires a particular ordering
of rows. Note that V_M lbc (MEac) = t, VM lbc (LIab ) = t and V>f lbc (06c) = /. So
(MEac —» (Llab -4 Oac )). That is, Ferio MLX is invalid.
This is the recipe for constructing a table X 2 for model MNxy (where x and y are
a, b or c) from a table T\ for model Mn (where r l\ has rows a, b and c): make row a
in Tj be row x in X 2 , make row c in T) be row y in T 2 , and make row b in T\ be the
third row in T 2 . Every row in To must be an a-row, a 6-row or a c-row. So, for example,
consider the Baroco/XMM cell on table 5, which is marked with ‘lab’. Use the recipe to
construct table 8 for model Mi a b, which invalidates Baroco XMM. (The a-row of table
6 becomes the a-row of table 8; the c-row of 6 becomes the b- row of table 8; and the
6-row of 6 becomes the c-row of table 8.) Since Mi a b(Acb) — t, Miab(MOab) = t and
Miab(MOac) = /, [A (Acb —4 (MOab -4 MOac )).
Aristotle's Modal Syllogisms
275
Table 8. Model Mi a b
n +
q + n
q
a
1
2
3
b
3
1,2
c
1 , 2,3
Model M 2 expressed by table 9 may be used to show that Celarent XLL is invalid.
Table 9. Model M .2
n +
q +
n
q~
a
1
2
3
b
1
2
3
c
2
3
1
Table 9 expresses a model. For all x and y , + x <£. n~ (y). So conditions B3 and B5 are
trivially satisfied. For all x and y if + (rr) o + (y) then n + (x) o + (y). So B4 is satisfied.
For all x and y, if + (a:) C n + (y) then n~(y) C n~(x). So B6 is satisfied.
Celarent XLL is invalid since: i) VM 2 (Ebc) = t since + (b) does not overlap + (c); ii)
VM 2 (LAab) = t since + (a) C n + (6); and iii) Vm 2 (LEcic) = / since + (a) g n“(c).
So VM 2 {Ebc -> ( LAab -4 LEac)) = /. So ( Ebc -4 ( LAab -4 LEac)).
Model Mz expressed by table 10 may be used to show that Camestres LXL is invalid.
Table 10. Model M$
n +
q +
n
q
a
1
2
3
b
3
2
1
c
3
2
1
Table 10 expresses a model. B3 and B5 are trivially satisfied since, for every x and y,
+ {x) ^ n~(y). For B4 note that if + (a;) o + (y) then n + (x) o + (y). For B6 note that if
+ (z) C n + (y) then n~(y) C n~(z).
Camestres LXL is invalid since: i) VM 3 (LAcb) = t since + (c) C n+(6); ii) Vm 3 {Eab)
t since + (a)fl + (6) = 0; and iii) Vm 3 {LEcic) — f since + (a) 2 n~(c). So VM 3 (LAcb -4
(Eab — > LEac ) = /. So ^ ( LAcb -4 ( Eab —> LEac)).
For Aristotle, Camestres LXL is invalid since Celarent XLL is invalid. A “semantic
rule” that underwrites this reduction of an invalidity to an invalidity may be stated as
follows.
276
Fred Johnson
R^-DR3 i) From (p -4 (q —»■ r)) and j= (p —> s) infer ^ (s —> (g —>■ r)); and ii)
from ^ (p —> (q —> r)) and |= (<7 -4 s) infer ^ (p —>■ (s —> r)).
Proof. For i) Suppose a) ^ (p —> (q -4 r)) and b) |= (p -4 s). By a) there is a
model M such that Vm(p) = i, V^{q) = i and Vm{t) = /■ By b) Vm(s) = f. So
^ (s —» (g -4 r)). Use similar reasoning for ii). ■
R^-AW is the semantic counterpart of the syntactic rule R-DR3, which is called re¬
jection by antecedent weakening (R-AW). Given the rejection of Celarent XLL ( Ebc —>
(LAab — > Llac )) and the conversion principle h ( Ebc -4 Ecb), Camestres LXL is re¬
jected by R-AW. Likewise, given the invalidity of Celarent XLL and the semantic conver¬
sion principle (= ( Ebc -4 Ecb), Camestres LXL is invalid by R^-AW.
Semantic counterparts of other syntactic rejection rules may be put to use to establish
invalidities. We illustrate this point by considering the semantic counterpart of R-RV.
R^-RV i) From (p -4 q) infer {->q -4 —>p); ii) from ^ (p -4 (9 -4 r)) infer V 1
(p -4 (t —» ->< 7 )); and iii) from ^ (p -4 {q -4 r)) infer ^ (~>r -4 (p -4
Proof. For i) suppose bMp -4 9 ). So there is a model M such that Vm{p ) = t and
Vm(?) = /- So VM(-<q) = t and Vm(~'P) = /• So ^ (->(/ -4 ->p). Use similar
reasoning for ii) and iii). ■
By R^-RV, since ( Ebc -4 {LAab LEac)) (Celarent XLL is invalid), ^
(fj&c -4 (-1 LEac -> -'LAab)). By using semantic counterparts of other syntactic prin¬
ciples stated above it is easy to conclude that Festino XMM is invalid.
To show that Baroco XLL is invalid we use model Ad 4, presented on table 11. 17
Table 11. Model Ad 4
n +
q +
71
9
a
1,2
3,4
b
1,3
4
2
c
1
4
3
2
Table 11 expresses a model. Base conditions B3 and B5 are trivially satisfied since, for
every x and y, + (a:) 2 n ~{y )• B4 and B6 are trivially satisfied since, for every x and y,
n + (x) ^ n+ (y)-
Given model 7 W 4 : i) Um 4 (Ac&) = t since + (c) C + {b)\ ii) VM 4 {LOab) = t
since n + (a) o n~b\ and iii) VM 4 {L0ac) = f since n + (a) does not overlap n~(c).
So V M4 ((Acb -4 ( LOab -4 LOac))) = f. So ^ ( Acb -4 ( LOab -4 LOac)). Fol¬
lowing the pattern indicated above we record on table 5 the invalidity of Baroco XLL by
17 Thomason [1993, p. 127] uses this table to invalidate Baroco XLL and Bocardo LXL though his definition
of “validity” is not identical to that which we are currently discussing. Thomason models are discussed below.
Aristotle’s Modal Syllogisms
277
putting ‘4ac’ in the Baroco/XLL cell, where ‘4’ refers to model M 4 and ‘ac’ indicates
that ‘a’ and ‘c’ are taken as minor and major terms, respectively.
Aristotle’s counterexample for Baroco XLL is controversial. According to Thom on p.
148 of [1991] Aristotle used terms ‘animal’, ‘man’ and ‘white’, generating the purported
counterexample: ‘All men are animals; some whites are necessarily not animals; so some
whites are necessarily not men.’ Thom says:
The problem with this counter-example is not (as van Rijen supposes [1989])
that the major premise is necessarily true. It is that, if the minor is taken to
be true then the conclusion will be true also.
In agreement with Thom, Aristotle did not provide a good counterexample for Baroco
XLL. A better informal counterexample is found in [Johnson, 1993, p. 179]: ‘All things
that are chewing are bears ( Acb)\ some animals (dogs, say) are necessarily not bears
( LOab ); so some animals are necessarily not chewing ( LOac )’. We do not follow Thom
in developing formal systems that take Baroco XLL to be invalid.
Though models Mi to M 4 and variants of them constructed by interchanging rows
in them suffice to give countermodels for the invalid syllogisms marked on table 5, other
models are needed to invalidate all of McCall’s rejection axioms and thus all of the invalid
wffs. The model used in [1989] to invalidate McCall’s (LAbb ->(LAff -*(Aad -KLAda
->(MAae -KLAcb -KLAbd -»(LAce -KAec -KLAfc ->(MAdf ->MAac))))))))))) (*5.41
on p. 59) has four members. It is presented on table 12.
Table 12. Model for *5.41
n +
q +
n q
a
1 , 2 , 3,4
b
3,4
1,2
c
4
3
1 2
d
3,4
1,2
e
3,4
1,2
f
4
2 1,3
An implication of [Johnson, 1989] is that every invalid L-X-M wff of form (pi
(p 2 —l > ... —J► p n ) ...), where each pi (for 1 < i < n) is a simple wff or the negation of
a simple wff, may be shown invalid by using a model (W ,...) in which W has no more
than 6 members. 18
In the next section we shall examine valuable attempts by Thomason to improve on the
semantics discussed in this section.
18 [Johnson, 1991] shows that W does not require more than 3 members if all simple sentences are assertoric
and all terms are “chained”.
278
Fred Johnson
5.1 Thomason models
In [Thomason, 1993] three notions of models are defined that enable Thomason to obtain
soundness and completeness results for McCall’s L-X-M calculus. In contrast to the
soundness and completeness proofs given in [Johnson, 1989] no use is made of rejection
axioms and rejection rules. One of these models comes close to the notion of a model
defined above. We call it a “T3-model” (his “models”) and define it as follows.
Definition 5.5. (T3-model) M is a T3-model iff M = (W,n + ,q + ,n~ ,q~), where W is
a non-empty set and n + , q + , n~, and q~ are functions that map terms into subsets of W
and satisfy the following “base conditions”, where + (x) is short for n + (x) U q + (x ):
B1 If f and g are any of the functions n + ,q + ,q~ or n~ and f ^ g, then, for every term
x, f(x) fl g(x) — 0; and for every x, n + (x) U q + {x) U q~{x) U n~(x) = W
B2 For every x, n + (x) ^ 0
BT3 (For every x and y) if + (x) o + (y) then + (x) o n + (y)
BT4 If + (x) C n~(y) then + (t/) C n~(x)
BT5 If + (x) C n + (y) then n~ (y) Cn~ (x)
To define “valuation relative to a model” and “validity” Thomason uses the same su-
perstructural conditions as used above.
Thomason, on p. 133 of [1997], says that in his [1993] he “tried to find simpler, and
apparently weaker, requirements for models” than those given in [Johnson, 1989]. In the
motivating section of [Thomason, 1993] he says “Johnson ... provided a semantics that
has the right validities, but the latter is in some sense contrived.” No doubt conditions
BT3, BT4 and BT5 are more easily understood than B3, B4, B5 and B6 but Thomason is
not correct in saying that the former, taken collectively, are weaker than the latter, taken
collectively. We use the following theorem to show the relationship between T3-models
and “J-models”, the models defined above that satisfy base conditions B1 to B6.
Theorem 5.6. i) Every T3-model is a J-model, but ii) there are J-models that are not
T3-models.
Proof. For i) suppose M is a T3-model. First, suppose + (z) C n~{y) and + (x) C + (y).
Then, by BT4, + {y) C n~{z) . Then + (x) C n~{z). Then M satisfies B3. Next,
suppose + (y) C n + (z) and + (x) o + (y). Then + {y) ° + (x) and, by BT3, + (y) o n + (x).
Then n + (x) o n + (z). Then M satisfies B4. Next, suppose + (y) C n~(z) and + (x) o
+ (y). Then, by BT3, n + (x) o n + ( 2 :). Then M satisfies B5. Next, suppose + (z) C n + (y)
and n + (x) o n~{y). Then, by BT5, n~(y) C n~(z). Then n + (x) o n~(z). Then M
satisfies B6.
For ii) note that Mi, specified in table 11, is a J-model but not a T3-model since condition
BT3 is not satisfied. Note that + (a) o +(b) but + (a) does not overlap n + (b). ■
Though both T3-models and J-models, with the superstructural conditions defined
above, will reveal the invalidity of any invalid syllogism with any finite number of an¬
tecedents (or premises), it is not clear that BT3 and BT5 are Aristotelian principles. Cer¬
tainly BT4 is Aristotelian, given 25a27-28. And J-models may be simplified by replacing
B3 with BT4, given the following theorem.
Aristotle’s Modal Syllogisms
279
Theorem 5.7. i) B3 is derivable from BT4; and ii) BT4 is derivable from B3.
Proof. For i) suppose that a) if + {z) C n~(y) then + (y) C n~(z) and that b) + (z) C
n~(y) and + (x) C + (y). Then n + (y) C n~(z). Then + (x) C n~(z).
For ii) suppose that c) if + (z) C n~(y) and + (x) C +(y) then + (x) C n~(z) and d)
+ (z) C n~{y). Since +(y) C+ (y), +(y) C n~{z). ■
B4 (Darii LXL), B5 (Ferio LXL) and B6 (Baroco LLL) are Aristotelian given 30a37-b2
and 30a6-14 of the Prior Analytics .
5.2 Variants of the L-X-M system
Paul Thom in [1991, p. 137] points out that condition B2, used in the definitions of J-
models and T3-models to guarantee that McCall’s axiom Llaa is valid, is unAristotelian.
He says that it is unAristotelian to think that there are walkers that are essentially walkers
and whites that are essentially white. Johnson’s [1993] and [1995] provide variants of
McCall’s L-X-M that are sound and complete systems, where condition B2 is omitted.
Both systems have 100% Aristotelicity. The systems deviate from McCall’s in that lines
in deductions need not be axioms or lines that are ultimately derived from axioms by
rules of inference. The systems are “natural deduction systems” rather than “axiomatic
systems”. Proofs of completeness assume that the inferences under discussion satisfy
what Smiley calls the “chain condition” in [Smiley, 1994, p. 27]. And the systems attempt
to accommodate Aristotle’s proofs by ecthesis . I9 In the remainder of this section we
illustrate proofs by ecthesis and then discuss the chain condition in the next section.
In addition to sentences such as Abe and Labe discussed above we count m £ a(m is
an a), m 6 n a (m is necessarily an a), m $ n a (m is necessarily not an a), etc. The latter
are singular sentences. In contrast to Thom’s view, to present proofs by ecthesis singular
sentences are required. 20 Consider this proof of Darapti XXX taken from Smith’s [1989,
p. 9] with my additions in square brackets:
When they [terms] are universal, then when both P [that is, c] and R [that is,
a] belong to every S [that is, b], it results of necessity that P will belong to
some R. ... It is ... possible to carry out the demonstration through ... the
setting out [that is, by ecthesis ]. For if both terms belong to every S, then if
some one of the S’s is chosen (for instance N [that is, m], then both P and R
will belong to this; consequently, P will belong to some R. (28al8-26)
Aristotle’s proof by ecthesis may be formalized as follows:
1. Abe (Premise)
2. Aba (Premise)
l9 The systems proposed by [Lukasiewicz, 1957], [Corcoran, 1972] and [Smiley, 1973] do not attempt to
accommodate Aristotle’s proofs by ecthesis . According to Thom's [Thom, 1991] account of ecthesis both
Baroco XLL and Bocardo LXL are valid, though Aristotle regarded them as invalid.
20 For an alternative method of working with singular sentences in the context of syllogistic reasoning see
[Johnson, 1999a]
280
Fred Johnson
3. m £ b (By ecthesis from 1. Since all b are c there must be a b that may be referred
to as m.)
4. m £ c (From 1 and 3. Since all b are c and m is a b it follows that m is a c.)
5. m £ a (From 2 and 3 by the reasoning for line 4.)
6. lac (From 4 and 5 by “Existential Generalization” - if a particular object m is both
an a and a c then something is both an a and a c.)
Aristotle proves that Baroco LLL is valid in the following passage, taken from Smith’s
[1989, p. 131:
... it is necessary for us to set out that part [m] to which each term [b and
c] does not belong and produce the deduction about this [m]. For it will be
necessary in application to each of these; and if it is necessary of what is set
out, then it will be necessary of some part [a] of the former term (for what is
set out is just a certain “that”. (30a9-15)
His proof by ecthesis may be formalized as follows:
1. LAcb (Premise. Whatever is c is necessarily b.)
2. LOab (Premise. There is something that is necessarily a but necessarily not b.)
3. m e n a
4. m $ n b (Lines 3 and 4 come from line 2 by ecthesis . This is a use of “Existential
Instantiation”.)
5. m c (From 1 and 4. If whatever is c is necessarily b and m is necessarily not in
c then m is necessarily not in c.)
6. LOac (From 3 and 5 by Existential Generalization.)
6 THE CHAIN CONDITION, RELEVANCE LOGIC AND THE AP SYSTEM
The following remarks by Smiley from two of his papers show that Aristotle held views
endorsed by contemporary “relevance logicians”. 21
By building onto the propositional calculus Lukasiewicz in effect
equates syllogistic implication with strict implication and thereby commits
himself to embracing the novel moods corresponding to such theorems as
21 It is very surprising that Aristotle is scarcely mentioned in [Anderson and Belnap, 1975 and 19921, which
provides authoritative discussions of relevance logic. See McCall’s discussion of “connexive implication” [1975
and 1992, pp. 434-452] for the one reference to Aristotle. In [Johnson, 1994] a syllogistic logic is developed that
is a “connexive logic”. Pleasing relevance logicians, the logic satisfies both Aristotle’s thesis (If y is the logical
consequence of a non-empty set of premises. A', then X is semantically consistent) and Boethius's Thesis (If z
is the logical consequence of a set of premises, X U y, then z is not the logical consequence of a set of premises
X U y\ where y' contradicts y). Ironically, neither Aristotle’s nor Boethius's thesis holds for what is now
known as the “classical propositional calculus”. In [Johnson, 1994] a theorem is proven that has as a corollary
this interesting result due to C. A. Meredith in [19531: The number of valid n-premised assertoric syllogisms
(for n > 2) is 3n 2 + 5rr + 2. There is no question that in Chapter 25 of Book I of the Prior Analytics Aristotle
was looking for such a general result. Given the chain condition such counts are possible.
Aristotle’s Modal Syllogisms
281
((Aab A Oab) -4 led) or ((Aab A Acd) -4 Aee ). On the other hand Aris¬
totle’s own omission of these syllogisms of strict implication, as they may
be called can hardly be written off as an oversight. For they violate his dic¬
tum that a syllogism relating this to that proceeds from premises which relate
this to that’ (41a6). This dictum is part of a principle which is absolutely
fundamental to his syllogistic, namely the principle that the premises of a
syllogism must form a chain of predications linking the terms of the conclu¬
sion. Thus his doctrine of the figures, which provides the framework for his
detailed investigation of syllogistic, is founded on this principle (40b30 ff.)
Not less important is that the chain principle is essential to the success of
his attempt at a completeness proof for the syllogistic. By this I mean his
attempt to show that every valid syllogistic inference, regardless of the num¬
ber of premises, can be carried out by means of a succession of two-premise
syllogisms. [Smiley, 1973, pp. 139-140]
Probably the easiest way to formulate this ‘chain condition’ is to use the
notation AB to denote any of the forms a, e, i, o regardless whether the
subject is A or B. Then the condition is that a valid argument must be of the
form ‘ AC, CD, DE, EF,... GH, HB\ therefore AB'. The chain condition
dramatically alters the character of the completeness problem (for a start,
it excludes the possibility of anything following from an infinite number of
premises) and it permits simple strategies for the proof that would otherwise
be inconceivable. It is therefore not surprising that Aristotle’s proof should
fail to fit the same picture as, for example, Corcoran’s own completeness
proof for syllogistic logic without the chain condition [Corcoran, 1972]. 22
[Smiley, 1994, p. 27]
Aristotle’s case for the chain condition is redolent of relevance — the need
for some overt connection of meaning between premises and conclusion as a
prerequisite for deduction. [Smiley, 1994. p. 30]
Since McCall’s presentation of the L-X-M calculus imitates Lukasiewicz’s, it also em¬
braces “novel moods” of the sort mentioned by Smiley. (LAab -4(-i LAab -4lcd)) is
asserted in L-X-M even though neither c nor d occurs in the antecedent (and thus the
consequent is irrelevant to the antecedents). This follows from the completeness result,
mentioned above, for L-X-M. Note that for every model M either Vm {LAab) = f or
^M^LAab — /• So |= ( LAab -A (->LAab — > Ide)). And ( LEab —» ( LEcd -4
( LEef -A- Igg))) is asserted in L-X-M. For in every model M, Vm{I99) = t. So
[= (LEab -A- (LEcd -4 ( LEef -4 Igg))). So, by completeness, h (LEab -4(LEcd
—KLEef ->Igg))) even though g does not occur in any of the antecedents.
22 Corcoran gives a Henkin-style completeness proof for the assertoric syllogistic. His system validates infer¬
ences such as ‘ Eab\ so Acc’, inferences eschewed by relevance logicians. This inference is valid for Corcoran
since the conclusion is logically true, even though the premise is irrelevant to the conclusion.
282
Fred Johnson
By using the chain condition in [1973], Smiley formulates an elegant decision proce¬
dure for the assertoric syllogistic. In [Johnson, 1994] a system is developed for Aristotle’s
apodeictic syllogisms, call it the “AP system”, that uses the chain condition. A decision
procedure is given for it that yields Smiley’s decision procedure as a corollary . 23 Both
decision procedures are given below.
Definition 6.1. (chain condition) Let Pr t - refer to “prefixes” of assertoric or apodeictic
sentences: A, E, I, O, LA, LE, LI, LO, MA, ME, MI and MO. A chain is a set of
sentences whose members can be arranged as a sequence (Pr\[x 1 X 2 ],
Pr 2 [x 2 X 3 ],..., Pr n [x n x\\), where Pr t [xiXj] is either PriX{Xj or PriXjXi and Xi 7 / Xj
if i ^ j.
So, for example, {LAab, MAcb, Lied, Ead] and {Oba, LEbc, LEdc, LAda) are
chains. But neither { LAaa } nor {LAab, Aba, MAac, Aca } is a chain.
Definition 6.2. (abbreviations for subsets of chains) X/LAxy refers to Axy or LAxy.
X/LAx - y refers to 0 if a: = j/; otherwise, it refers to {XjLAz\z 2 , X/LAz 2 z%,...,
X/LAz n ~\z n }, a subset of a chain, where x = z\,y = z n and n > 1. LAx - y refers
to 0 if x = y, otherwise it refers to X/LAx - z, LAzy, a subset of a chain. X/LExy
refers to Exy or LExy. X/LIxy refers to Ixy or LIxy. And X/LOxy refers to Oxy or
LOxy.
So, for example, LAab, LAbe, Acd has form X/LAa - d, but does not have form
LAa — d. LAab, LAbc, Acd, LAd — e has form X/LAa — e and form LAa — e.
Definition 6.3. (contradictory of, cd) Let cd(Axy) — Oxy where ‘cd’ may be read as
‘the contradictory of’. Let cd(Ixy) = Exy, cd{LAxy) = MOxy, cd{LExy) = MIxy,
cd{LIxy) = MExy, and cd{LOxy) = MAxy. And let cd(cd(x)) = x. So, for exam¬
ple, cd(Exy) = Ixy.
Theorem 6.4. (Johnson [1994], decision procedure for “AP-validity”) Suppose “valid,tp”
(apodeictic syllogistic validity) is defined as in [1994]. Consider an inference in the “AP
system” from premises P\,P 2 ,... ,P n to conclusion C. This inference is valid/ip if and
only if {P\,P 2 ,..., P n , cd(C)} is a chain that has one of the following eleven forms:
1. X/LAx-y, X/LOxy
2. LAx-z, MAzu, LAu-y, LOxy
3. X/LAx-z, LAzy, MOxy
4. X/LAz-x, X/LAz-y, X/LExy
5. X/LAz-x, X/LAz-u, MAuv, X/LAv-y, LExy (or LEyx)
6 . X/LAz-x, X/LAz-u, LAuy, MExy (or MEyx)
7. X/LAz-x, X/LAu-y, X/LIzu, X/LExy (or X/LEyx)
8 . X/LAz-x, X/LAu-v, MAvw, X/LAw-y, X/LIzu (or X/LIuz), LExy (or LEyx)
9. X/LAz-x, X/LAu-y, MIzu, LExy (or LEyx)
10. LIxy, MExy (or MEyx)
23 See [Johnson, 1994] and [Johnson, 1997] for other systems that yield Smiley’s decision procedure as a
special case of a more general decision procedure.
Aristotle’s Modal Syllogisms
283
11. X/LAz-x, X/LAu-v, LAvy, X/LIzu (or X/LIuz), MExy (or MEyx)
So, for example, { LAab, LAbc , Acd, cd(Aad)} has form 1. So ‘ LAab, LAbc, Acd; so
Aad' is valid. {LAab, LAbc, Acd, cd(MAad)} has form 1. So ‘ LAab, LAbc, Acd; so
MAad ’ is valid. {Aab, cd(Obc), Acd, Oad } has form 1. So ‘ Aab, Acd, Oad; so Obc is
valid. {MAac, LAcb, cd(MAab)} has form 2. So "MAac, LAcb; so MAab ’ (Barbara
LMM) is valid. {LAcb, lac, cd(LIab)} has form 11. So "LAcb, lac; so Llab’ (Darii
LXL) is valid.
Notice that since ‘E’ occurs at most once in any of the forms, it follows that no valid
syllogism, regardless of the number of premises, is such that ‘E’ occurs in two or more of
its premises. A similar comment applies to occurrences of ‘M’.
The following result is a corollary of theorem 6.4.
Theorem 6.5. (Smiley [1973], decision procedure for “AS-validity”) Suppose “valid^”
(assertoric syllogistic validity) is defined as in [1973]. Consider an inference in the asser-
toric syllogistic from premises Pi, P 2 ,..., P n to conclusion C. This inference is valid^g
if and only if {Pi, P 2 , . .., P„, cd(C )} is a chain that has one of the following three forms:
1. Ax-y, Oxy (restriction of form 1 of theorem 6.4)
2. Az-x, Az-y, Exy (restriction of form 4 of theorem 6.4)
3. XAz-x, Au-y, Izu, Exy (or Eyx) (restriction of form 7 of theorem 6.4)
So, for example, given form 2 of the corollary both ‘ Aca, Acb; so lab’ (Darapti) and
‘Aca, Eab; so Ocb’ (Celaront) are valid.
On table 13 a syllogism is marked as valid by referring by number to the form listed in
theorem 6.4 in virtue of which it is valid. So, for example, the first occurrence of ‘1’ on
the table indicates that Barbara XXX, XXM, XLX, XLM. LXX, LXM, LLX and LLM are
valid in virtue of their relationship to {X/LAx — y, X/LOxy}. The 333 valid syllogisms
marked on the table exactly match the 333 syllogisms that McCall accepts in his L-X-M
system. Seep. 46 of [McCall, 1963].
7 CONTINGENT SYLLOGISMS
A. N. Prior [1962, p. 188] gives a simple account of “the usual meaning of ‘contingent’”
in the following passage:
In the De Interpretatione Aristotle remarks that the word ‘possible’ is am¬
biguous; we should sometimes say that ‘It is possible that p ’ follows from
‘It is necessary that p', but sometimes that it is inconsistent with it. In the
former sense ‘possible’ means simply ‘not impossible’; in the latter sense,
‘neither impossible nor necessary’. It is for ‘possible’ in this second sense
that the word ‘contingent’ is generally used. That is, ‘It is contingent that
p ’ means ‘Both p and not-p are possible’, KMpMNp [or (Mp A M->p)].
Contingency in this sense stands between necessity and impossibility, but in
quite a different way from that in which the simply factual stands between the
284
Fred Johnson
Table 13. Valid^p 2-premised syllogisms
X/L
L
L
X
L
M
M
X
L
M
XJL
L
X
L
M
L
X
M
M
L
X/M
L
L
L
M
M
M
M
X
X
Barbara
1
3
3
2
2
Celarent
7
9
9
8
11
8
11
Darii
7
11
11
9
8
8
9
Ferio
7
8
8
9
11
11
9
Cesare
7
9
9
8
11
8
11
Camestres
7
9
9
11
8
11
8
Festino
7
8
8
9
11
11
9
Baroco
1
2
3
2
3
Darapti
4
6
6
6
5
5
5
5
Felapton
4
5
5
5
6
6
5
Disarms
7
11
11
8
9
9
8
Datisi
7
11
11
9
8
8
9
Bocardo
1
2
2
3
3
Ferison
7
8
8
9
11
11
9
Bramantip
4
6
6
5
5
5
5
Camenes
7
9
9
11
8
11
8
Dimaris
7
11
11
8
9
9
8
Fresison
7
8
8
9
11
11
9
Fesapo
4
5
5
5
6
6
5
Barbari
4
6
6
5
5
5
5
Celaront
4
5
5
5
6
6
5
6
Cesaro
4
5
5
5
6
6
5
6
Camestrop
4
5
5
6
5
6
5
Camenop
4
5
5
6
5
6
5
Total
8x24
24
15
8
24
24
16
7
15
8 = 333
necessary and the possible. It is not that necessity implies contingency, and
contingency impossibility; rather we have three mutually exclusive alterna¬
tives which divide the field between them — either a proposition is necessary,
or it is neither-necessary-nor-impossible (i.e. contingent), or it is impossible
On p. 190 of [ 1962] Prior introduces the symbol ‘Q’ and reads ‘Qp’ as ‘It is contingent
that p\ McCall adopts Prior’s use of ‘Q’ to refer to Aristotle’s contingency operator and
Thom [1994, p. 91] refers to [McCall, 1963] to support his use of ‘Q’ in his discussions
of contingency. In the discussion below, we shall also use ‘Q’. 24
24 The following symbols are also found in the literature that formalizes contingency: ‘E' 2 [Becker-Freyseng,
1933], ’T’ [Lukasiewicz, 1957] and ‘P()’ [Smith, 19891. Smith's P(Aab) is McCall's QAab, and Smith's PAab
is McCall’s MAab. Lukasiewicz used ‘T’ instead of Q’ since earlier in his book he used Q' for ‘is equivalent
to’. McCall’s Barbara LQM is Ross’s [1949] A n A c A p . Montgomery and Routley use V for contingency in
[1966] and [1968]. And Cresswell uses V for contingency and A for non-contingency in [1988]
Aristotle's Modal Syllogisms
285
Thom makes the following remarks about contingency at the beginning of his article
(P- 91):
Aristotle’s contingency syllogistic deals with the logic of derivations involv¬
ing propositions that contain an expressed mode of contingency. The contin¬
gent is defined at I. 13, 32“ 18-20, as that which is not necessary, but which
being supposed does not result in anything impossible, i.e. as two-sided pos¬
sibility.
Fitting Prior’s remarks, the two sides of contingency (Q) are necessity and impossibil¬
ity. The one side of possibility (M) is impossibility.
McCall in [1963] diminishes and extends the L-X-M calculus, formulating the Q-L-X-
M calculus. We give the basis for it.
Primitive symbols
Use the primitive symbols for L-X-M together with
monadic operator Q
Formation rules
Use the formation rules for L-X-M, amending FR2' as follows.
FR2' If p is a categorical expression then ->p is a categorical expression and Lp and Qp
are wjfs.
Assertion axioms
Use A0-A4 from system LA and A5-A14 from system L-X-M. So A2 is IAA. Add the
following axioms.
A15 (Barbara QQQ)
A16 (Darii QQQ)
All (QXQ-AAE, figure 1)
A18 (Darii QXQ)
A19 (Barbara XQM)
A20 (Celarent XQM)
A21 (Ferio XQM)
A22 (complementary conversion, QE-QA)
A23 (complementary conversion, QI-QO)
A24 (complementary conversion, QO-QI)
A25 (QI conversion)
A26 (QE-ME subordination)
A27 (QI-MI subordination)
KQAbc -KQAab —»QAac))
KQAbc -KQIab ->QIac))
KQAbc ->(Aab —»QEac))
KQAbc -»(Iab -*QIac))
h (Abc —KQAab ->MAac))
h (Ebc -KQAab -s-MEac))
h (Ebc —KQIab ->MOac))
KQEab —>QAab)
KQIab —>QOab)
h (QOab ->QIab)
h (QIab ->QIba)
h (QEab —>MEab)
KQIab —»MIab)
286
Fred Johnson
A28 (QO-MO subordination) h (QOab ->MOab)
Assertion transformation rules
Use the assertion tranformation rules for L-X-M.
On p. 76 of [1963] McCall gives the following reason for changing A2 from Llaa to
Iaa.
If we retain the axiom Llaa, we may, by means of the substitution
CKQAacLIaalac [(( QAac A Llaa) —A lac))) of Darii QLX (proved be¬
low), derive the implication CQAacIac [(QAac —» 7oc)], which is un-
Aristotelian.
We shall present this reasoning systematically.
Proof.
1. h (Ebc -A ( QAab —> MEac)) (by A20)
2. h (QAab —A ( Ebc —A MEac)) (from 1 by AI)
3. h (QAab -A ( Llac -A Ibc)) (from 2 by RV and SE)
4. h (QAab —> ( Llaa -A Iba)) (from 2 by US)
5. h Llaa (by A2 for L-X-M)
6. H ( QAab -A Iba) (from 4 and 5 by AI and MP)
7. h (QAab -A lab) (from 6 by CW, given Con)
8. h (QAac -A lac) (from 7 by US) ■
McCall devised his system Q-L-X-M so that it has this feature: (QEab -A QEba)
is not accepted. He wishes to reflect Aristotle’s view that universally negative contin¬
gent propositions are not convertible. 25 McCall puts Aristotle’s argument for the non¬
convertibility of such propositions as follows:
... in 36b35-37a3, Aristotle gives what is in essence the following argument.
We know that QAab implies QEab, and that QEba implies QAba [by com¬
plementary conversion]. Therefore if QEab implied QEba, QAab would
imply Q Aba, which it does not. Hence QEab is not convertible.
But, unfortunately, McCall’s Q-L-X-M system is too strong. It forces us, for example,
to accept ( QAbc -A ( LAab -A LAde)), which is clearly unAristotelian. It does not
satisfy the chain condition mentioned above. After showing this, we shall lay out a system
that is semantically consistent and maximizes Aristotelicity.
25 0n p. 198 of [1957] Lukasiewicz calls Aristotle’s view a ‘grave mistake'. Lukasiewicz says ‘He [Aris¬
totle] does not draw the right consequences from his definition of contingency, and denies the convertibil¬
ity of universally-negative contingent propositions, though it is obviously admissible.’ But, following Mc¬
Call, one can attempt to formulate Aristotle’s contingency syllogistic without, in effect, defining QEab as
(^LEab A -^L-^Eab).
Aristotle’s Modal Syllogisms
287
7.1 Overlooked acceptances in the Q-L-X-M system
McCall claims that Barbara QLX is not a thesis in his Q-L-X-M system. See table 13 on
p. 92 of [1963]. But this result is a corollary of the following theorem.
Theorem 7.1. H (Q Abe -»(LAab — >x)), where x is any wff.
Proof.
1. h (Eca —> (QAbc —> MEba)) (by A20 and US)
2. h ( LAab -4 LIba) (by Ap-sub-a)
3. '-(MEba -4 MOab ) (from 2 by RV and SE)
4. h (Eca -4 (QAbc -4 MOab)) (from 1 and 3 by CW)
5. h (QAbc —» (LAab -4 7ca)) (from 4 by AI, RV and SE)
6. h (LAab —> (lea -4 LIcb)) (A7 and US)
7. h ((QA6e -4 (LAab -4 Tea)) -4 ((LAab -4 (7ca -4 LIcb)) -4 (QAbc -4
(LAab -4 LIcb)))) (by AO)
8. l ~(QA6c -4 (LAab -4 LIcb)) (from 5,6 and 7 by MP)
9. h (QA6c -4 Q£6c) (by CC and US)
10. h (QEbc -4 MEbc) (by A26 and US)
11. h (MEbc -4 M Ecb) (by Ap-con and US)
12. h (Q,46c -4 MEcb) (from 9, 10 and 11 by CW)
13. h (QA6c -4 -.ZJcb) (from 12 by SE)
14. h ((Q.46c -4 (LAab -4 L7d>)) -4 ((QA&c -4 -ZJcb) -4 (QAbc -4 (LAo6 -4
a;)))) (by AO)
15. h (QAbc -4(LAab -4x)) (from 8, 13, and 14 by MP) ■
The following theorem provides additional evidence that McCall’s Q-L-X-M system is
too strong to be Aristotelian.
Theorem 7.2. l ~(LAbc ->(QAab 4i)), where x is any sentence.
Proof.
1. h (Eac -4 (Qlba -4 MObc)) (by A21 and US)
2. h (QAab -4 Qlba) (by A18, US, A2, MP)
3. h (Eac -4 ( QAab -4 MObc)) (from 1 and 2 by AS)
4. h (LAbc —> (QAab -4 iac)) (from 3 by RV and SE)
5. '-(lac —» 7ca) (by Con)
6. h (LAbc -4 (QAab -4 7ca)) (from 4 and 5 by CW)
7. '-(QAab -4 (7ca -4 Qlcb)) (by A18)
8 . h ((LAbc —> (QAab -4 7ca)) -4 ((QAab -4 (Ica -4 Qlcb)) -4 (LAbc -4
(QAab -4 Qlcb)))) (by AO)
9. ''(LAbc -4 (QAafr —> Qlcb)) (from 6,7 and 8 by MP)
10. l_ (Q7c6 -4 Qlbc) (by A25 and US)
11. h (Qlbc -4 QObc) (by A23 and US)
288
Fred Johnson
12. h (QObc -4 MObc) (by A28 and US)
13. H ( LAbc -4 ( QAab -4 MObc)) (from 9, 10, 11 and 12 by MP)
14. h (LAbc -4 ( QAab —> -i LAbc )) (from 13 by SE)
15. h ((LA&c —» ( QAab —> -> LAbc)) -4 (LAbc -4 (QAafe -4 a:))) (by AO)
16. h (LAbc —KQAab -4a:)) (from 14 and 15 by MP) ■
According to McCall’s table 13 on p. 92 of [McCall, 1963], sentences representing
Barbara QLX, Barbara LQX, Barbara LQQ, Baroco QXM and Bocardo XQM are not
accepted in the Q-L-X-M system, though they correspond to inferences that Aristotle
considered to be valid. But it is an immediate consequence of theorems 7.1 and 7.2 that
the first three sentences are accepted. That the last two are accepted may be seen as
follows:
1. h (LAbc -4 (QAab -4 Aac)) (by theorem 7.2)
2. h (Oac —> (QAab — > MObc)) (from 1 by RV and SE)
3. h (Obc —» (QAba -4 MOac)) (Bocardo XQM, from 2 by US)
4. h (QAbc -4 (LAab —> Aac)) (by theorem 7.1 )
5. h (QAbc -4 (Oac -4 MOab)) (from 4 by RV and SE)
6. h (QAcb —» (Oab -4 MOac)) (Baroco QXM, from 5 by US)
So McCall’s claim on p. 93 of [1963] that Q-L-X-M has 85% Aristotelicity needs to be
modified. Instead of 24 “non-Aristotelian moods” out of 154 moods marked on his table
13, there are 29 out of 154. So the Aristotelicity of the Q-L-X-M system is about 81%.
When determining the Aristotelicity of a system, McCall only uses figures 1, 2 and
3 and none of the “subaltern moods” such as Barbari. Given theorems 7.1 and 7.2, the
following wffs are accepted in Q-L-X-M, though they are not marked as accepted on
McCall’s table 13: Bramantip QLQ, Camenes LQQ, Fesapo QLQ and Barbari LQQ.
In the following section we modify Q-L-X-M so that the resulting system, QLXM',
does not have the unAristotelian features that result from theorems 7.1 and 7.2. Given
the data - that Aristotle regarded Barbara LQM as invalid and Bocardo QLM as valid, for
example - it is a virtue of the modified system that it does not have 100% Aristotelicity.
Note that if h (QObc —> (LAba -4 MOac)) (Bocardo QLM) then *" (LAac -4 (QObc -4
MOba)) (Baroco LQM) by Reversal. In system QLXM' both Barbara LQM and Bocardo
QLM are invalid. In contrast, in system Q-L-X-M both are valid.
8 QLXM'
To ensure that theorems 7.2 and 7.1 may not be proven in system QLXM' we exclude
axioms A20 (Celarent XQM) and A21 (Ferio XQM). This decision is not difficult to
make since, as McCall points out, Aristotle’s proofs of Celarent XQM and Ferio XQM
are flawed. McCall shows that one who endorses such reasoning, thinking that “what is
impossible cannot follow from what is merely false, but not impossible”, is committed to
the absurd consequence that ‘Some B are A; all C are A; so some C are A’ is valid.
Aristotle’s Modal Syllogisms
289
Only one other Q-L-X-M axiom is excluded to form QLXM': delete axiom A28,
h (QOab —> MOab). This decision is a result of semantic considerations. For A28 to
be validity preserving QOab and LAab must be semantically inconsistent. Since LAab
is true iff + (a) C n + (b), we could make QOab and LAab contraries by fixing the se¬
mantics so that QOab is true iff + (a) o q(h). But then we are forced to say that Bocardo
QLQ, for example, is valid even though Aristotle considered it to be invalid. (Suppose
that VMiQOab) = t and Vm(LAgc ) = t. Then + (a) o q(b) and + (a ) C n + (c). Then
n + (c) o q(b). Then + (c) o q(b). Then Vm(QOc6) = t.) Note, also, that if QOab is
true iff + (a) o q(b) then we would want to ensure that Qlab is true iff + (a) o q(b) to
guarantee the soundness of the complementary conversion principles that Aristotle clearly
supported. But then we would be forced to say that Disamis QLQ is valid even though
Aristotle considered it to be invalid. (Suppose that V>f(Q/6c) = t and V/^iLAba) = t.
Then + (b) o q(c) and + (b) C n + (a). Then n + (a) ° q(c). Then + (a) o q(c). Then
Vm{QIo,c) = t.) Similar remarks may be made about Disamis QXQ, Datisi LQQ and
Datisi XQQ.
Rather than fixing the conditions for the truth of QOab as indicated above we may
let QOab be true iff either Qlab or Qlba is true. 26 To make A28 truth preserving we
must ensure that if LAab is true then both Qlab and Qlba are false. Such a position
does not fit the sorts of examples Aristotle uses. Suppose, for example, that all things that
are sleeping are necessarily men. It does not follow that it is not true that some men are
contingently sleeping.
We avoid the above difficulties by deleting axiom A28 when defining QLXM'.
In this system, as in Q-L-X-M, there are no rejection axioms and no rejection rules.
Before giving a semantics for QLXM' we shall establish some immediate inferences
that are conversions, subalternations or subordinations. With them we shall show the
acceptance of various two-premised syllogisms indicated on table 15 by leaving a cell
unmarked. After the semantics is given we shall show that sentences corresponding to the
other cells, those in which numerals occur, are invalid. An occurrence of the “hat sign”
in a cell in the table means the entry conflicts with Aristotle’s judgments about validity as
recorded on McCall’s authoritative table 12 of [McCall, 1963]. 27
Theorem 8.1. (Ordinary Q-conversions, Q-con) i) h ( Qlab -A Qlba) ; and ii) h ( QOab —>
QOba).
Proof, i) is A25. For ii) use A23, A24 and CW. ■
Theorem 8.2. (Contingency subalternations, Q-sub-a) i) h (QAab —> Qlab); ii) ( QAab —>
QOab); iii) h (QEab -4 Qlab); and iv) h (QEab —4 QOab).
Proof. For i) use A18, AI, A2 and MP. For ii) use i), A23 and CW. For iii) use i), A22
and AS. For iv) use ii), A22 and AS. ■
26 Thom evaluates QOab in this way in [1993] and [1994],
27 In the notes for table 12 McCall comments on tables in [Becker-Freyseng, 1933, p. 88] and [Ross, 1949.
after p. 2861.
290
Fred Johnson
Table 14. McCall’s Table 12 and RV inconsistencies
Q
Q
X
Q
L
Q
X
Q
L
Q
L
Q
X
Q
L
Q
X
Q
L
Q
L
Q
Q
Q
Q
Q
Q
M
M
X
X
M
M
Barbara
V
V
V
V
v 13
V
Celarent
V
V
V
V 2
V 7
V
yl4
V
Darii
V
V
V
V
V s
4
12
V
V
Ferio
V
V
V
V 3
V 9
V
v 15
V
Cesare
v lu
3
V
15
V
Camestres
V 4
V
6
V
17
Festino
V 11
2
V
14
V
Baroco
1
5
13
16
Darapti
V
V
V
V
V
V
V
Felapton
V
V
V
V
V
V
V
V
Disamis
V
V
V
V
V
11
7
V
V
Datisi
V
V
V
V
V
10
9
V
V
Bocardo
V
V 5
V
V 16
V
Ferison
V
V
V
V 6
V 12
8
V 17
McCall follows Ross’s use of “complementary conversion” to refer to A22 to A24.
On p. 298 of [Ross, 1949] Ross, in his discussion of 35a29-bl, identifies the following
entailments, endorsed by Aristotle, as “complementary conversions”:
‘For all B, being A is contingent’ [QAba] entails ‘For all B, not being A
is contingent’ [QEba] and ‘For some B, not being A is contingent’ [QOba].
’For all B , not being A is contingent’ [QEba] entails ’For all B, being A is
contingent’ [QAba] and ‘For some B, being A is contingent’ [Qlba]. ‘For
some B, being A is contingent’ [Qlba] entails ‘For some B , not being A is
contingent’ [QOba]. ‘For some B, not being A is contingent’ [QOba] entails
‘For some B , being A is contingent’ [Qlba].
Given the following theorem and US, Ross’s six complementary conversions are as¬
serted in QLXM'.
Theorem 8.3. (Complementary conversion, CC) i) h (QAab —> QEab) \ ii) h (QAab -4
QOab)\ iii) h (QEab -4 QAab)\ iv) h (QEab -4 Qlab); v) h (QIab —> QOab)\ and vi)
h (QOab -4 Qlab).
Proof. For i) use A17, US, AI, A1 and MR For ii) use Q-sub-a, A23 and CW. iii) is A22.
For iv) use iii), Q-sub-a and CW. v) is A23. vi) is A24. ■
Theorem 8.4. (Complementary conversions per accidens , CC(pa)) i) h {Q Aab —> Qlba );
ii) h (QAab -4 QOba)\ iii) h (QEab -4 Qlba ); and iv) h (QEab —> QOba).
Aristotle’s Modal Syllogisms
291
Table 15. QLXM' countermodels
Q
Q
X/L
Q
X
Q
L
Q
L
Q
XJL
Q
X
Q
L
Q
L
Q
Q
Q
Q
M
M
X
X
M
M
Barbara
5ac
7ab
8ac
Celarent
6ac
lac
Darii
5ac
7ab
8ac
Ferio
6ac
lac
5cb
Cesare
7ac
9ca
6ac
5ca
lac
Camestres
7ac
6ca
9ac
lac
5ac
Festino
7ac
9ca
6ac
lac
Baroco
7ac
6ca
9ac
lac
1 lbc
1 lbc
1 lbc
Darapti
7cb
7bc
Felapton
9bc
8 be
lbc
Disamis
5ca
7cb
7bc
Datisi
5ca
7cb
7bc
Bocardo
5ca
9bc
11 ac
8bc
1 lac
llac
Ferison
9bc
8 be
5cb
Bramantip
5ca
8ca
7ba
Camenes
lOac
6c a
7bc
lac
Dimaris
5ca
8ca
7ba
Fresison
7ac
5ca
6bc
8bc
Fesapo
5ca
6bc
8bc
Barbari
5ac
7ab
8ac
Celaront
6ac
lac
Cesaro
7ac
9ca
6ac
lac
Camestrop
7ac
6ca
9ac
lac
lab
Camenop
6ca
lac
Proof. For i) use Q-sub-a, Q-con, US and CW. For ii) use i), A23, US and CW. For iii)
use i), A22 and AS. For iv) use iii), A23, US and CW. ■
Theorem 8.5. (Contingency subordinations, Q-sub-o) i) h (QAab —> MAab)\
ii) h (QEab —> MEab)\ and iii) h (QIab —» Mlab).
Proof. For i) use A19, Al, US and MP. ii) is A26. iii) is A27. ■
Uses of AS or CW in proofs of the following theorems involve only those immediate
inferences that have been proven above. So, for example, in the proof that Celarent QQQ
is asserted AS is used with Q-sub-a and US ( h (QEbc -» QAbc)) and CW is used with
Q-sub-a and US (~(QAac —> QEac)).
Theorem 8.6. (asserted QQQs) The non-numbered QQQ cells on table 15 correspond to
asserted sentences.
292
Fred Johnson
Proof.
1. h (QAbc —KQAab —jQAac)) (Barbara QQQ, by A17)
2. h (QEbc —KQAab -4QEac)) (Celarent QQQ, from 1 by AS, US, CW)
3. h (QAbc —KQIab —»QIac)) (Darii QQQ, by A16)
4. h (QEbc —KQIab ->QOac)) (Ferio QQQ, from 3 by AS, US, CW)
5. h (QAbc -KQAba —>-QIac)) (Darapti QQQ, from 3 by AS, US)
6. h (QEbc —KQAba ->QOac)) (Felapton QQQ, from 5 by AS, US, CW)
7. h (QAbc —KQIba —»QIac)) (Datisi QQQ, from 3 by AS, US)
8. h (QEbc —KQIba ->QOac)) (Ferison QQQ, from 7 by AS, US, CW)
9. h (QIbc —KQAba ->QIac)) (Disamis QQQ, from 7 by AI, CW, US)
10. h (QObc —KQAba ->QOac)) (Bocardo QQQ, from 9 by AS, US, CW)
11. h (QIcb —KQAba —»QIac)) (Dimaris QQQ, from 9 by AS, US)
12. h (QAcb —KQAba —jQIac)) (Bramantip QQQ, from 11 by AS, US)
13. h (QEcb —KQAba ->QOac)) (Fesapo QQQ, from 12 by AS, US, CW)
14. h (QAbc —KQAab ->QAac)) (Barbari QQQ, from 1 by CW, US)
15. h (QEbc —KQAab ->QEac)) (Celaront QQQ, from 2 by CW, US)
16. h (QAcb -KQEba —►QOac)) (Camenop QQQ, from 12 by AS, US, CW)
Theorem 8.7. (asserted QXQs and XQQs) The non-numbered QXQ and XQQ cells
table 15 correspond to asserted sentences.
Proof.
1. h (QAbc —KAab ->QAac)) (Barbara QXQ, by A17, US, CW)
2. l ~(QEbc —KAab ->QEac)) (Celarent QXQ, from 1 by AS, US, CW)
3. h (QAbc —Klab ->QIac)) (Darii QXQ, by A18)
4. h (QEbc ->(Iab —>QOac)) (Ferio QXQ, from 3 by AS, US, CW)
5. h (QAbc —KIba ->QIac)) (Datisi QXQ, from 3 by AS, US)
6. h (QEbc —^(Iba -^QOac)) (Ferison QXQ, from 5 by AS, US, CW)
7. h (QAbc —KAba —>QIac)) (Darapti QXQ, from 5 by AS, US)
8. l ”(QEbc —KAba —^QOac)) (Felapton QXQ, from 7 by AS, US, CW)
9. h (QAbc —KAab ->QIac)) (Barbari QXQ, from 1 by US, CW)
10. h (QEbc —KAab —>QOac)) (Celaront QXQ, from 2 by US, CW)
11. h (Ibc -KQAba —>QIac)) (Disamis XQQ, from 5 by AI, CW, US)
12. h (Abc —KQAba ->QIac)) (Darapti XQQ, from 11 by AS)
13. h (Icb —KQAba -*QIac)) (Dimaris XQQ, from 11 by AS)
14. h (Acb —KQAba ->QIac)) (Bramantip XQQ, from 13 by AS)
15. h (Acb -KQEba ->QOca)) (Camenop XQQ, from 14 by AS, CW)
Theorem 8.8. (asserted QLQs and LQQs) The non-numbered QLQ and LQQ cells
table 15 correspond to asserted sentences.
Proof. Use theorem 8.7 and Sub-o.
Aristotle’s Modal Syllogisms
293
So, for example, h (QAbc ->(LAab —>QAac)) since h (QAbc —KAab —»QAac)) by the¬
orem 8.7 and since '“(LAab —»Aab) by Sub-o.
Theorem 8.9. (asserted QXMs and XQMs) The non-numbered QXM and XQM cells on
table 15 correspond to asserted sentences.
Proof. For non-numbered QXM and XQM cells referred to by names that do not end
with ‘o’ use theorem 8.7 wherever possible with Q-sub-o and CW. So Barbara QXM
is asserted since Barbara QXQ is assserted. And, by this reasoning, Celarent QXM,
Darii QXM, Darapti QXM, Datisi QXM, Barbari QXM, Darapti XQM, Disamis XQM,
Bramantip XQM, Camenes XQM, Dimaris XQM and Barbari XQM. For the remaining
non-numbered cells use asserted MXM syllogisms from table 13 wherever possibile with
Q-sub-o and AS. So, Ferio QXM is accepted since Ferio MXM is accepted. And, by
this reasoning, Festino QXM, Felapton QXM, Disamis QXM, Ferison QXM, Bramantip
QXM, Dimaris QXM, Fresison QXM, Fesapo QXM, Celaront QXM, Cesaro QXM, Darii
XQM, Datisi XQM and Barbari XQM. The only remaining non-numbered QXM and
XQM cells correspond to the axiom Barbara XQM (A21) and Camenop XQM, which is
deduced from Camenes XQM by CW given Ap-sub-a. ■
Theorem 8.10. (asserted QLXs and LQXs) The non-numbered QLX and LQX cells on
table 15 correspond to asserted sentences.
In the following proof the asterisks mark inconsistencies in the data as reported on
McCall’s table 12 on pp. 84-85 of [ 1963].
Proof. Use theorem 8.9 with RV and SE. So i) Celarent QLX is asserted since Festino
QXM is asserted; ii) Celaront QLX (is asserted) since Cesaro QXM (is asserted); iii)* Ce-
sare QLX since Ferio QXM; iv) Camestres QLX since Darii QXM; v)* Festino QLX since
Celarent QXM; vi)* Baroco QLX since Barbara QXM; vii) Cesaro QLX since Celaront
QXM; viii) Camestrop QLX since Barbari QXM; ix) Camenes QLX since Dimaris XQM;
x) Fresison QLX since Camenes XQM; xi) Fesapo QLX since Camenop XQM; xii) Ca¬
menop QLX since Bramantip XQM; xiii) Celarent LQX since Disamis XQM; xiv) Ferio
LQX since Datisi XQM; xv) Celaront LQX since Barbari XQM; xvi) Cesare LQX since
Datisi QXM; xvii)* Camestres LQX since Ferison QXM; xviii) Festino LQX since Dis¬
amis QXM; xix) Cesaro LQX since Darapti QXM; xx) Camestrop LQX since Felapton
QXM; xxi) Felapton LQX since Barbari XQM; xxii) Bocardo LQX since Barbara XQM;
xxiii)* Ferison LQX since Darii XQM; xxiv) Camenes LQX since Fresison QXM; xxv)
Fresison LQX since Dimaris QXM; xxvi) Fesapo LQX since Bramantip QXM; and xxvii)
Camenop LQX since Fesapo QXM. ■
Theorem 8.11. The non-numbered QLM and LQM cells on table 15 correspond to as¬
serted sentences.
Proof. For the QLMs use: i) results for the accepted QXM syllogisms stated in theorem
8.9, Sub-o and AS; or ii) results for the accepted QLX syllogisms stated in theorem 8.10,
Sub-o and CW. So, for example, h (QAbc —» (LAab -> MAac)) (Barbara QLM is as¬
serted) since h (QAbc —> (Aab —» MAac)) and h (LAab —» Aab) given AS. h (QEcb —>
294
Fred Johnson
(LAab —4 MEac)) (Cesare QLM is asserted) since h (QEcb —4 (LAab —4 Eac)) and
h (Eac —4 MEac).
For the LQMs use results for the XQMs in theorem 8.9 or the LQXs in theorem 8.10
together with Sub-o, AS or CW. So, for example, h (LAbc —4 (Qlab —4 Mlac )) (Darii
LQM is asserted) since h (LAbc —4 (Qlab —4 Jae)) and h (/oc —4 Mlac). h (LEbc —4
(Qlab —4 MOac)) (Ferio LQM is asserted) since h (LEbc -4 (Qlab —4 Oac)) and
h (Ooc —4 MOac)). m
Theorem 8.12. The non-numbered QQMs on table 15 correspond to asserted sentences.
Proof. Obtain the assertion of Barbara QQM from the assertion of Barbara QQQ by using
CW with Q-sub-o. Use similar reasoning for Celarent, Darii, Barbari, Darapti, Disamis,
Datisi, Bramantip and Dimaris. We generate the remaining four QQMs as follows.
1. ^ (QEbc —4 (QAab —4 QEac)) (Celarent QQQ)
2. h (QEac —4 MEac) (by Q-sub-o)
3. h (MEac —4 MOac) (by Ap-sub-a)
4. h (QEac -4 MOac) (from 2 and 3 by CW)
5. h (QEbc -4 (QAab -4 MOac)) (Celaront QQM, from 1 and 4 by CW)
6. h (QAab -4 (QEbc -4 QEac)) (from 1 by AI)
7. h (MEac -4 MEca) (by Ap-con)
8. h (QEac -4 MEca) (from 2 and 7 by CW)
9. h (QAab -4 (QEbc -4 MEca)) (from 6 and 8 by CW)
10. h (QAcb —4 (QEba —4 MEac)) (Camenes QQM, from 9 by US)
11. h (MEac —4 MOac) (Ap-sub-a and US)
12. h (QAcb -4 (QEba —4 MOac)) (Camenop QQM, from 10 and 11 by CW)
13. h (QEcb —4 (QAba —4 MOac)) (Fesapo QQM, from 12 by CC and AS) ■
8.1 Semantics for QLXM'
The semantics for QLXM' is given by referring to Q-models.
Definition 8.13. (Q-model) M is a Q-model iff M = (W, n + ,q + ,n~ ,q~), where W is
a non-empty set and n + ,q + ,n~, and q~ are functions that map terms into subsets of W
and satisfy the following “base conditions”:
BQ1 If / and g are any of the functions n + ,q + ,q or n and f ^ g, then, for every
term x , f(x) fl g(x) = 0; and for every x , n + (a:) U q + (x) U q~ (x) U n~ (x) = W
BQ2 (For every x and y) if + (x) C n~(y) then + (y) C n~(z)
BQ3 If + (y) C n + (z) and + (x)o + (y) thenn + (x)o n + (z)
BQ4 If + (y)^ n (z) and + (x) o + (y) then n + (x) o n (z)
BQ5 If + (.c) C n + (y) and n + (x) o n (y) then n + (x) o n (z)
BQ6 If + (y) C q(z) and + (x) C q(y) then + (x) C q(z)
BQ7 If + (?/) ^ q(z) and + (x) o q(y) or q(x) o + (y) then + (x) o q(z) or q(z) o + (x)
Aristotle’s Modal Syllogisms
295
BQ8 If + (j/) C +(z) and + (a;) C q(y) thenn + (:r) does not overlap n (z)
Definition 8.14. (valuation relative to a Q-model) Vm > s a valuation relative to a Q-
model .Miff it is is a valuation that satisfies the following “superstructural conditions”:
51 (For every x and y) Vm (Axy) = t iff + ( x ) C + ( y )
52 V M {Ixy) = t iff + (z) o +(y).
53 V M (LAxy) = tiff+ (x) C n+(y)
54 VM(LIxy) = t iff n + (x) o n + (y)
55 VM(L-'Axy) = t iff n + (x) o n~(y)
56 VM(L-'Ixy) = t iff + (x) C n~(y)
57 V M {QAxy) = t iff + (x) C q(y)
58 V M (QIxy) =t 'tff + (x) ° q{y) or q{x) o +(y)
59 V M (Q~'Axy) = t iff + (z) o q(y) or q(x) o + (y)
sio v M (Q-'ixy) = t iff+(*) c q(y)
Definition 8.15. (Q-valid) |=q a (a is Q-valid) iff, for every Q-model M,
Vyvi(a) = t. a is Q-invalid (|=q q) iff a is not Q-valid.
Theorem 8.16. (soundness) If a is an assertion in QLXM' then (=q a.
Proof. We need to show that i) if h a is an axiom of QLXM' then \=q q; and ii) each
assertion transformation rule of QLXM' preserves Q-validity. Some examples of the rea¬
soning needed are given. For Al, (=q Aaa since, for every Q-model M, VM(Aaa) = t
since + (o) C + (a). For A2, (=q Iaa since, for every Q-model M, VVt(/oo) = t
since + (a ) o + (a). For A5, suppose there is a Q-model M such that VM(LAbc) = t ,
V M (Aab) = t and V M (LAac ) = /. Then +(&) C n+(c), + (a) C +(6) and + (a ) £
n + (c), which is impossible. So [=q (LAbc (Aab —> LAac )). For A15, suppose
there is a Q-model M such that VM(QAbc) = t , V M {QAab) = t and Vm{QAo.c) = f.
Then + (6) C q(c), + (a) C q(b) and + (o) £ q(c), which is impossible given BQ6.
So [=q (QAbc —a (QAab -> QAac)). For ArI suppose \=q (... x ... x ...) but
(... y ... y ...), where (... y ... y ...) is the result of replacing every occurrence of term
x in (... x... x ...) with term y. Then, for some Q-model M , Vm (... y ... y ...) = /,
where Vm(v) is set S. Let Vm (x) = S. Then V M (- ■ ■ x ... x ...) = /. So it is impossible
for ArI not to preserve validity. For Ar 2 suppose a) |=q (p q), b) |=q p and c) \£q q.
Then, for some Q-model M, VM{q) = /, given c). Then, by a), Vm{p) — /. which
conflicts with b). So Ar 2 preserves validity. Reasoning for the other axioms and rules is
straightforward and is omitted. ■
Given the soundness of QLXM' every asserted sentence in Lukasiewicz’s LA is Q-
valid since LA is a fragment of QLXM'. All of the L-X-M syllogisms marked as asserted
on table 5 are Q-valid since all of them are asserted in QLXM'. And, given the following
theorem, all of the syllogisms marked as invalid on table 5 are Q-invalid.
Theorem 8.17. Models M\, M 2 , M 3 and M\ are Q-models.
296
Fred Johnson
Proof. By earlier arguments the four models satisfy conditions BQ1 to BQ5. Consider
M\. Suppose + (y) C q(z). Then z — c. So BQ 6 is trivially satisfied. For all x,
+ (x) o q[c ). So BQ7 is satisfied. For all z, n~(z) — 0. So BQ 8 is satisfied. Consider
M. 1 - Suppose + (y ) C q{z). Then y = a or y = b, and 2 : = c. So BQ 6 is trivially
satisfied. For all x , + (x) o q(c). SoBQ7 is satisfied. For all z, if + (c) C + (z) thenz = c.
Since n~(c) = 0, BQ 8 is satisfied. Consider _A4 3 . Suppose + (y) C q(z). Then y = b or
y = c, and z = a. So BQ 6 is trivially satisfied. For all x, + (a)oq(x). So BQ7 is satisfied.
For all z, if + (o) C + ( z ) then z = a. Since n~ (o) = 0, BQ 8 is satisfied. Consider M. 4 .
For all y and z, if + (y) g! g(z). So BQ 6 , BQ7 and BQ 8 are trivially satisfied. ■
Table 16. Q-model M 5
n +
Q +
n
Q
a
1
2
3
b
3
1
2
c
1,3
2
Table 16 expresses a model. BQ1 and BQ2, here and below, require no comment.
For every y and if +(y) C n + (z) then z = c. For every x, n + {x) o n + (c). So
BQ3 is satisfied. For every y and z, + (y) Q. n~(z). So BQ4 is trivially satisfied. If
+ {y) Q n + (z) then z = c. For every x, y and z, if z C n + (y) then n + (z) does not
overlap n~(y). So BQ5 is satisfied. For every x and y, if x C q(y ) then x — a and y = b.
So BQ6 is trivially satisfied. For all x, x o q(b). So BQ7 is satisfied. For all z, + (a) does
not overlap n~(z). So BQ8 is satisfied.
Given Q-model AI 5 , (LAbc —> (QAab —> LAac)). For, VM 5 {LAbc) — t since
+ ( 6 ) C n + (c). VM 5 (QAab) = t since + (a) C q(b). And VM 5 {QAac) = / since
+ ( a ) 2 q( c )- The occurrence of ‘4ac’ in the Barbara/LQQ cell indicates Q-model M 5 is
a countermodel for Barbara LQQ, where ‘a’ is the minor term and ‘c’ is the major term.
This method of listing minor and major terms will be followed below.
Table 17. Model M e
n +
q +
n
Q
a
1
2
3,4
b
3
1
4
2
c
4
1 , 2,3
Table 17 expresses a model. For every y and z, if + (y) C n + (z) then y = z = c. For
every x and y, if + (x) o + (y) then x — c. Since n + (c) o n + (c), BQ3 is satisfied. For
every y, if + (y) C n~(a) then y = b or y = c. For every x, if x o + (b) orio + (c)
then n + (x) o n~(a). For every y, if + (y) C n~(b ) then y = b. For every x, if x o +(6)
Aristotle’s Modal Syllogisms
297
then n + (x) o n~(b). For every y, if + (y ) C n~(c) then y — a or y = b. For every
x, if x ° + (a) or x o + (b) then n + (x) o n~(c). So B4 is satisfied. For every y and
z, if + (z) C n + (y) then n~(y) C n~(z) (that is, Thomason’s BT5 is satisfied). 28 So,
BQ5 is satisfied. For every x and y, if + (x) C q(y) then x = a and q = b. So BQ 6 is
trivially satisfied. For all w, if + (w) o q(a) or q(w) o +(o) then w = a or w = b. Since
+ (a) o q(b) and + ( 6 ) o 9 ( 6 ), BQ7 is satisfied. For every z, if + (b) C z then z = b. Since
+ (a) does not overlap n~(b), BQ 8 is satisfied.
Given Q-model Me, not \=q ( LEac —» ( QAab -* QObc)). For, Vm 6 {LEcic) = t
since + (a) C n~(c). V_M 6 (QAab) = t since + (a) C q(b). And Vm 6 (QObc) = / since
+ (b) does not overlap q(c ) and q(b) does not overlap + (c). The occurrence of ‘6bc’ in
the Felapton/LQQ cell indicates that Q-model Me is a countermodel for Felapton LQQ,
where ‘b’ is the minor term and ‘c’ is the major term.
Table 18. Model M 7
n +
q+
n
Q
a
1
3
2
b
2
3
1
c
1
3
2
Table 18 expresses a Q-model. Since Thomason’s BT3 (if + (x) o + (y) then + (x) o
n + (y)) is satisfied, both BQ3 and BQ4 are satisfied. Since BT5 is satisfied BQ5 is
satisfied. If + (x) C q{y) then y = b and either x — a or x = c. Then BQ 6 is trivially
satisfied. If + (z) o q(a) or q(z) o + (a) and if + (z) o q(c) or q(z) o +(c) then + (z) o +(&).
So BQ7 is satisfied. If + (6) C + (z) then z = b. Since n~(b) = 0, BQ 8 is satisfied.
Use Q-model Ads to show that Barbari LQX and others are invalid.
Table 19. Model Ms
n +
q +
n
q
a
1
2
3
4
b
4
3
1,2
c
3,4
2
1
Table 19 expresses a Q-model. Suppose + (t/) C n + (z). Then y = b or y = c, and
z — c. Since n + (b) o n + (c) and n + (c ) o n + (c), BQ3 is satisfied. Since there is no
x such that n + (x) o n _ (c), BQ5 is satisfied. Since, for every x and y, x y, BQ4 is
28 As noted above, BQ5 is a weaker condition than BT5. Replacing BQ5 with BT5 in the definition of a
Q-model, forming a Q' model, yields this highly unAristoteiian result: \=q, ( QAab -> Eab). For, suppose
that for some Q'-model M, V M (Eab) = /. Then V M (Iab) = t. By S2 and BT5. +(a) o n + (b). By S9,
V M (QAab) = f.
298
Fred Johnson
trivially satisfied. Suppose + (y) C q(z). Then y = a and z = b. Then BQ6 is trivially
satisfied. Since, for every a;, + (x ) °q(b) or q(x) o + (b), BQ7 is satisfied. If + (b) C + (z)
then z = 6 or z = c. Since, for all x, n + (x) does not overlap n~(b) and n + (x) does not
overlap n~(c), BQ8 is satisfied.
Table 20. Model Ad 9
n +
q +
n
Q
a
1
2
3,4
b
3,4
1
2
c
3
1 , 2,4
Table 20 expresses a Q-model. If + (y) C n + (z) then y = c and either z = c or z — b.
If + (re) o +( c ) then x = b or x = c. Since n + (x) o n + (z), BQ3 is satisfied. For every
y , if + (y) C n~(a) then y = c. For every x, if x o +(c) then n + (x) o n~(a). There
are no y such that y C n~(b). For every y, if + {y) C n~(c) then y = a. For every x,
if x o +(o) then n + (x) o n~(c). So BQ4 is satisfied. If + (z) C n + (y) then z = c and
either y = b or y — c. n + (x) does not overlap n~(b). If n + (x) o n~(c) then x = a or
x = b. Since n + (a) o n - (c) and n + (b) o n“(c), BQ5 is satisfied. For every x andy, if
x C q(y) then x — a and y = b. So BQ6 is trivially satisfied. For all z, if + (z) o + (a)
then z = a or z = b. Since + (a) o q(b) and + (b) ° q{b), BQ7 is satisfied. For all z, if
+ (b) C + (z) then b = z. Since n~(z) = 0, BQ8 is satisfied.
Table 21. Model M 10
n +
n
9 "
a
1
2
3,4
b
3
2
i
4
c
4
2
1,3
Table 21 expresses a Q-model. For every x and y, + (x) 2 n+ {y)- So BQ3 and BQ5
are satisfied. For all x and y, + (x) 2 n ~(y)■ So BQ4 is satisfied. If + (x) C q(y) and
+ (y) C q(z) then x = a and z = c. So BQ6 is satisfied. For every x and y, + (x) o q(y)
or q(x) o + (y). So BQ7 is satisfied. If + (x) C q(y) then x — b or x = c. For all z, n + (b)
does not overlap n~(z) and n + (c) does not overlap n~(z). So BQ8 is satisfied.
Table 22 expresses a Q-model. Since BT3 is satisfied, BQ3 and BQ4 are satisfied.
Since BT5 is satisfied, BQ5 is satisfied. For every x and y, x y. So BQ6, BQ7 and
BQ8 are trivially satisfied.
Aristotle’s Modal Syllogisms
299
Table 22. Model M n
8.2 Q-valid moods needed for completeness
Aristotle did not discuss any moods with possiblity, as opposed to contingency, premises
(or antecedents). But, given the semantics proposed for QLXM' we must recognize the
Q-validity of some moods in which an M-wff is a premise (or an antecedent). In particular
Darii QMQ is Q-valid. So, to move in the direction of obtaining completeness results for
QLXM' we shall amend the system by making Darii QMQ axiom 29 (A29).
Theorem 8.18. (soundness of amended QLXM') Suppose QLXM' is amended by mak¬
ing the assertion of Darii QMQ, h (QAbc —> ( Mlab -» QIac)) be an axiom. Leave
everything else unchanged. Then the resulting system is sound.
Proof. Suppose M is a Q-model, V^^QAbc) = t and Vm(M lab) = t. Given the
definition of a Q-model, at least one of these three conditions is met: i) + (a) o + (i>), ii)
+ (a) o q(b) or iii) + (a) C n~(b). If i)is met then + (a) o q(c) and thus Vm ( QIac ) = t.
If ii) is met then + (a) o q(c) or q{a) ° + (c) and thus Vm{QI(ic) = t. If iii) is met
then V^iMIab) = t and Vm(M lab) = /. Given this absurdity Vm{QIo.c) = t. So
[=q (QAbc —> (Mlab QIac)). ■
Assertions that are Q-valid correspond to unmarked cells on table 23. The marks in
cells indicate how countermodels may be found for the Q-invalid syllogisms the table
refers to.
For each unmarked cell we shall show how the indicated syllogism is asserted in the
system.
Theorem 8.19. (asserted QMQs and MQQs) The non-numbered QMQ and MQQ cells
on table 23 correspond to asserted wffs.
Proof.
1. h (QAbc ( Mlab ->• QIac)) (Darii QMQ, A29)
2. h (QEbc —> ( Mlab —>■ QOac)) (Ferio QMQ, from 1 by CC, AS, CW)
3. h (QAbc -> ( MIba -* QIac)) (Datisi QMQ, from 1 by Ap-con, AS)
4. h (QAbc -> ( MAba — » QIac)) (Darapti QMQ, from 3 by Ap-con, AS)
5. h (QEbc —> ( MIba —» QOac)) (Ferison QMQ, from 3 by CC, AS, CW)
6 . h (QEbc -> ( MAba —> QOac)) (Felapton QMQ, from 4 by CC, AS, CW)
7. h (QAbc —> (MAab —> QIac)) (Barbari QMQ, from 1 by Ap-sub-a, AS)
300
Fred Johnson
Table 23. Additional Q-syllogisms
QMQ
MQQ
QMM
MQM
QLL
LQL
QQM
Figure 1
Barbara
12ac
5ac
12ac
15ca
7ab
8ac
Celarent
12ac
6ac
13ac
7ac
7ab
Darii
5ac
15ca
7ab
Mbc
Ferio
6ac
Mac
Mac
7ab
15ba
Mac
Figure 2
Cesare
9ca
6ac
7ac
7ac
Mab
7ac
Camestres
6ca
9ac
7ac
7 ac
Mba
7ac
Festino
9ca
6ac
7ac
7ac
13ab
15ba
7ac
Baroco
6ca
9ac
7ac
7ac
12ab
Mba
7ac
Figure 3
Darapti
7cb
7bc
Felapton
9bc
Mac
Mac
7cb
15bc
Mac
Disamis
5ca
15ac
7cb
7bc
Datisi
5ca
15ca
7cb
Mbc
Bocardo
5ca
9bc
Mac
8bc
7cb
15ba
Mac
Ferison
9bc
Mac
Mac
7cb
15ba
Mac
Figure 4
Bramantip
5ca
15ac
8ca
7ba
Camenes
6ca
7bc
7ac
13ca
7ba
Dimaris
5ca
15ac
7cb
7ba
Fresison
5ca
6bc
7ac
7ac
Bab
15ba
7ca
Fesapo
5ca
6bc
7ac
8bc
16cb
15ba
Subalterns
Barbari
5ac
15ac
7ab
Mbc
Celaront
6ac
13ac
7ac
7ab
Cesaro
9ca
6ac
7ac
7ac
Bab
7ac
Camestrop
6c a
9ac
7ac
7ac
Mba
7ac
Camenop
6ca
7ac
16ac
7ba
8 . *~(QEbc —F (MAab —> QOac)) (Celaront QMQ, from 7 by CC, AS, CW)
9. h (MIbc — > (QAba —> QIac )) (Disamis MQQ, from 3 by AI, Q-con, CW)
10. h (MAbc —> ( QAba —> QIac)) (Darapti MQQ, from 3 by Ap-sub-a, AS)
11. h (MIcb -> (QAba —> QIac)) (Dimaris MQQ, from 1 by AI, Q-con, CW, US)
12. h (MAcb -» (QAba — > QIac)) (Bramantip MQQ, from 11 by Ap-sub-a, AS)
13. h (MAcb -> (QEba —> QOac)) (CamenopMQQ, from 12 by CC, AS, CW)
Theorem 8.20. (asserted QMMs and MQMs) The non-numbered QMM and MQM cells
on table 23 correspond to asserted sentences.
Proof. Use theorem 8.19 and Q-sub-o. ■
Theorem 8.21. (asserted QLLs and LQLs) The non-numbered QLL and LQL cells on
table 23 correspond to asserted wffs.
Aristotle’s Modal Syllogisms
301
Proof. Use theorem 8.20 and RV. ■
Theorem 8.22. (asserted QQMs) The non-numbered QQM cells on table 23 correspond
to asserted wffs.
Proof. Use theorem 8.6, Q-sub-o and CW for cells other than Camenes, Fesapo, Celaront
and Camenop QQM. For them use the following reasoning.
1. h {QEbc -4 (QAab -a MEac )) (Celarent QQM)
2. h (QEbc —» (QAab -4 MOac )) (Celaront QQM, from 1 by Ap-sub-a, CW)
3. h ( QAcb —► (QAba -a MEac)) (Camenes QQM, from 1 by AI, Ap-con, CW, US)
4. h (QAcb -4 (QAba -4 MOac)) (Camenop QQM, from 3 by Ap-sub-a, CW)
5. h (QEcb -4 (QA&a -4 MOac)) (Fesapo QQM, from 4 by CC, AS) ■
Table 24. Model M 12
n +
q +
n
q
a
i
2
3
4
b
3
4
2
i
c
2
3
1
4
Table 24 expresses a Q-model. For every x and y, + (x) n + (y). So BQ3 and BQ5
are trivially satisfied. For every x and y, + (x) <Z n~(y). So BQ4 is trivially satisfied.
Suppose + (y) C q(z). Then y = b and z — c. So BQ6 is trivially satisfied. For every x ,
+ (x) o q(b) or q(x) o + (b). So BQ7 is satisfied. If + (c) C z then z = c. Since n + (b)
does not overlap n~(c), BQ8 is satisfied.
Table 25. Model M 13
q +
n
q
a
i
2
3
4
b
4
3
2
1
c
1,2
3
4
Table 25 expresses a Q-model. Suppose + {y) C n + (z). Then y — a and z = c.
If + (x) o + (a) then x = a or x = c. Since n + (a) o n + (c), BQ3 is satisfied. Since
n~(c) — 0, BQ5is trivially satisfied. For every x and y, + (a;) ^ n ~(y )• So BQ4 is
trivially satisfied. Suppose + (?/) C q(z). Then y — b and z = c. So BQ6 is trivially
satisfied. For all x, + (c) o q(x). So BQ7 is satisfied. Since n~(z) = 0, BQ8 is satisfied.
Table 26 expresses a Q-model. For every x and y , if + (a:) C n + (y) then x = a and
y = c. If + (x) o + (a) then n + (x) o n + (c). So BQ3 is satisfied. For every x and y.
302
Fred Johnson
Table 26. Model Mu
n +
q + n
Q
a
i
2
3,4
b
3
4 1
2
c
1,2
4
3
+ (a;) £ n~(y). So BQ4 is satisfied. Since n~(c) = 0, BQ5 is satisfied. For every x and
y, if + (a:) C q(y) then y = a or y = c. So BQ6 is trivially satisfied. For every z, z o q(a)
or q(z) o + (a). And for every z, z o q(c) or q(z) o + (c). So BQ7 is satisfied. For every
z, if + (a) C + (z) then z = a. And for every z, if + (c) C + (z) then z = c. Since
n~(a) — 0 and n~(a) = 0 , BQ8 is satisfied.
Table 27. Model Mi 5
n +
q +
n
9
a
i
2,3
4
b
4
3
1,2
c
2
1,4
3
Table 27 expresses a Q-model. Since BT3 is satisfied, both BQ3 and BQ4 are satisfied.
Since BT5 is satisfied BQ5 is satisfied. Suppose + {y) C q(z). Then z = b. So BQ6 is
trivially satisfied. For all x, x o q(b). So BQ7 is satisfied. If + (b) C + (x) then x — b.
Since n~(b) — 0, BQ8 is satisfied.
Table 28. Model M\§
n +
q + n
9“
a
i
2
3
b
2
3
1
c
1,2
3
Table 28 expresses a Q-model. Suppose + {y) C n + (z). Then z = c. Since, for all
x, n + (x) o n + (c), BQ3 is satisfied. Since n~(c) — 0, BQ5 is satisfied. Since, for all x
and y, + (a;) 2 n ~{y)i BQ4 is trivially satisfied. Suppose + {y) C q(z). Then y = b and
z = a. So BQ6 is trivially satisfied. For all x, x o q(a). So BQ7 is satisfied. For all x,
n~(x) = 0. So BQ8 is trivially satisfied.
Note that the acceptance of all but two of the QLM and LQM moods is generated from
acceptances involving the MLM and LMM moods.
Aristotle’s Modal Syllogisms
303
9 THE ARISTOTELICITY OF QLXM'
Of the 154 first, second or third figure syllogisms referred to on table 15 there are exactly
thirteen that are Q-valid but invalid for Aristotle. And there are exactly nine that are
Q-invalid but are valid for Aristotle. So the Aristotelicity of QLXM' system is about
86 %. Of the twenty-two discrepancies seventeen are due to mistakes involving the use of
Reversal. These mistakes are marked on table 14 by using pairs of numbers from 1 to 17.
So, for example, on this table both Barbara QXM and Baroco QLX are marked with ‘1’,
indicating that by Reversal both should be valid or both should be invalid. But Aristotle
regarded only the former as valid. Both Ferison QLM and Camestres LQM are marked
with ‘17’, indicating that by Reversal both should be valid or both should be invalid.
Aristotle regarded only the former as valid.
So, there are five remaining discrepancies to account for. i) Darapti XQQ: Aristotle
could have used Disamis XQQ to show its validity, ii) Darapti LQQ: Aristotle could have
used Darapti XQQ to show its validity, iii) Festino QXM: As noted by McCall in [1963,
p. 93], Aristotle could have used Festino MXM to show its validity. Given Reversal,
Festino MXM is valid in virtue of Disamis XLL. iv) Celarent QLX: Aristotle could have
used Reversal and Festino QXM to show it is valid, v) Felapton XQM: Aristotle properly
regarded it as valid since he regarded Ferio XQM as valid. Given our interest in devel¬
oping a formal system that would not have the unAristotelian results, noted in theorem
7.2, which are present in McCall’s Q-L-X-M system, we chose to regard Ferio XQM as
Q-invalid.
10 TALLY OF THE TWO-PREMISED Q-VALID SYLLOGISMS
The 333 syllogisms marked on Table 13 are the Q-valid apodeictic two-premised syllo¬
gisms in which no contingent wff is a premise or a conclusion. Table 15 and table 23 refer
to some of the Q-valid 2-premised syllogisms that involve contingent wffs. To count all
of them we need to take account of complementary conversions. Note, for example, that
AEA QQQ-figure 1 (that is (QAbc -A (QEab -A QAac))) is Q-valid by complemen¬
tary conversion since Barbara QQQ is Q-valid. 29 In this section we shall count all of the
2-premised syllogisms that are Q-valid.
When counting the valid moods we shall use ‘[A]’ to mean that the premise or conclu¬
sion indicated may be either an A or an E wff. Similarly we shall use ‘[I]’ to mean the
premise or the conclusion indicated may be either an I or an 0 formula. So, by saying
that QQQ [A][A][A] in figure 1 is Q-valid, we are claiming the validity of eight figure 1
QQQ syllogisms: QQQ AAA (AAE, AEA, AEE, EAA, EAE, EEA, and EEE). By saying
that QXQ [A]I[I] in figure 1 is Q-valid, we are claiming the Q-validity of four figure I
QXQ syllogisms: QXQ All (AIO, Eli, and EIO).
29 See Ross’s table in [1949, facing p. 286] for references to this as well as several other syllogisms that may
be validated by using complementary conversion.
304
Fred Johnson
Q-valid QQQs (64):
Figure 1: [A][A][A], [A][I][I], [A][A][I]
Figure 3: [A][A]U], [I][A][I], [Aj[I][I]
Figure 4: [A][A][I], [I][A][I]
Q-valid QXQs and QLQs (40):
Figure 1: [A]A[A], [A]I[I], [A]A[I],
Figure 3: [A]A[I], [A]I[I]
Q-valid XQQs and LQQs (32):
Figure 3: A[A][I], I[A)[I]
Figure 4: A[A][I], I[A][I]
Q-valid QXMs (34):
Figure 1: [A]AA, [A]AE, [A]II, [A]IO, [A]AI, [A]AO
Figure 2: [A]IO, [A]AO
Figure 3: [A]AI, [A]AO, [I]AI, [A]II, [A]IO
Figure 4: [A]AI, [IJAI, [A]IO, [A]AO
Q-valid XQMs (20):
Figure 1: A[A]A, A[I]I, A[A]I
Figure 3: A[A]I,I[A]I, A[I]I,
Figure 4: A[A]I, A[A]E, I[A]I, A[A]0
Q-valid QLXs (24):
Figure 1: [A]AE, [A] AO
Figure 2: [A]AE, [A]EE, [A]10, [AJOO, (A]AO, [A]EO
Figure 4: [A]EE, [A]IO, [A]AO, JAJEO
Q-valid LQXs (30):
Figure 1: E[A]E, E[IJO, E[A]0
Figure 2: E[A]E, A[A]E, E[I]0, E[A]0, A[AJO
Figure 3: E[AJO, 0[A]0, E[I]0
Figure 4: A[A]E, E[I]0, E[A]0, A[A]0
Q-valid QLMs (46):
Figure 1: [A]AA, [A]AE, [AJII, [A]IO, [A]AI, [AjAO
Figure 2: [A]AE, [A]EE, [A]IO, [AJOO, [A]AO, [A]EO
Figure 3: [A]AI, [A]AO, [I]AI, [A]II, [A]IO
Figure 4: [A]AI, [A]EE, [I]AI, [A]IO, [AJAO, [AJEO
Q-valid LQMs (46):
Figure 1: A[A]A, E[A]E, A[I]I, E[IJO, A(A]I, E[AJO
Figure 2: E[A]E, A[A]E, E[IJO, E[AJO, A[AJO
Aristotle’s Modal Syllogisms
305
Figure 3: A[A]I, E[A]0,1[A]I, A[I]I, 0[A]0, E[I]0
Figure 4: A[A]I, A[A]E, I[A}I, E[I]0, E[A]0, A[A]0
Q-valid QMQs (16):
Figure 1: [A]I[I], [A]A[I]
Figure 3: [A]A[I], [A]I[I]
Q-valid MQQs (16):
Figure 3: A[A][I], I[A][I]
Figure 4: A[A][I], I[A][I]
Q-valid QMMs (8):
Figure 1: [A]II, [A]AI
Figure 3: [A]AI, [A]II
Q-valid MQMs (8):
Figure 3: A[A]I, I[A]I
Figure 4: A[A]I, I[A]I
Q-valid QLLs (8):
Figure 2: [A]EE, [A]EO
Figure 4: [A]EE, [A]EO
Q-valid LQLs (8):
Figure 1: E[A]E, E[A]0
Figure 2: E[A]E, E[A)0
Q-valid QQMs (48):
Figure 1: [A][A]A, [A][A]E, [A][I]I, [A][A]I, [A][A]0
Figure 3: [A][A]I, [I](A]I, [A][I]I
Figure 4: [A][A]I, [A][A]E, [I][A]I, [A][A]0
There are 333 + 64 + 40 + 32 + 34 + 20 + 24 + 30 + 46 + 46 +16 +16 + 8 + 8
+ 8 + 8 + 48 (that is, 781) Q-valid 2-premised syllogisms found in thirty five “general
moods”: LLX, LLM, LXX, LXM, XLX, XLM, XXX, XXM, LLL, LXL, XLL, LMM,
MLM, MXM, XMM, LMX, MLX, QQQ, QXQ, QLQ, XQQ, LQQ, QXM, XQM, QLX,
LQX, QLM, LQM, QMQ, MQQ, QMM, MQM, QLL, LQL, and QQM.
11 EXTENSIONS
The most natural extension of the above work on QLXM' would be to develop a Smiley-
type decision procedure for validity for the n-premised syllogisms, for n > 2, where these
syllogisms meet the chain condition. Though Smiley’s decision procedure for the asser-
toric syllogistic pairs inconsistent sets of wffs with syllogisms construed as inferences,
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Fred Johnson
the pairing could also be between sets of wffs and syllogisms constructed as implications.
The decision procedure would list Q-inconsistent sets such as {PiAx\X 2 , P 2 AX 3 X 4 ,...,
PnAx 2 n-iX 2 n, QAxix 2n }, where: i) each P iy for 1 < i < n, is X, L or Q\ ii) P n is Q\
and iii) Q (the negation of Q) is a new quantifier. Given the decision procedure it would
follow that ( QAab —> (Abe — k (LAcd —>■ (QAde —> QAae)))), for example, is Q-valid.
Though it is argued above that QLXM' is more Aristotelian than McCall’s Q-L-X-M
there are several other systems that could be developed to bring coherence into Aristotle’s
discussions of modalities. For example, consider Barbara XQM. McCall points out that
Aristotle’s defense of its validity is flawed, but McCall chooses to take it as an axiom in
his Q-L-X-M. It is also an axiom in QLXM'. Dropping this axiom would mean that the
semantics for the weaker system would be simpler.
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Logic of Relevance and Necessity, volume I and volume II. Princeton University Press, Princeton, 1975 and
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schlilsse: eine logischphilologische Untersuchung der Kapitel 13-22 von Aristoteles Analytica priora I.
Junker and Diinnhaupt, Berlin, 1933.
[Bell etal., 2001] John L. Bell, David DeVidi, and Graham Solomon. Logical Options: an Introduction to
Classical and Alternative Logics. Broadview, Peterborough, 2001.
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702, 1972.
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Dordrecht, 1998.
[Frege, 1960] Gottlob Frege. Negation. In Peter Geach and Max Black, editors, Translations from the Philo¬
sophical Writings of Gottlob Frege, pages 117-135. Blackwell, Oxford, second edition, 1960.
[Geach, 1964] P. T. Geach. Review of McCall, 1963. Ratio, 6:200-206, 1964,
[Geach, 1972] P. T. Geach. Logic Matters. Blackwell, Oxford, 1972.
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London, 1968.
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ledge, London, 1996.
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284, 1989.
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187, 1991.
[Johnson, 1993] Fred Johnson. Modal ecthesis. History and Philosophy of Logic, 14:171-182, 1993.
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422, 1994.
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Philosophy of Logic, 16:1-18, 1995.
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1997.
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[Lear, 1980] Jonathan Lear. Aristotle and Logical Theory. Cambridge University Press, Cambridge, 1980.
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matics. North-Holland, Amsterdam, 1963.
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bases for normal modal logics. Logique et Analyse, 35-36:318-328, 1966.
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1990.
[Patterson, 1989] Richard Patterson. The case of the two Barbaras: basic approaches to Aristotle's modal logic.
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and Philosophy of Logic, 11:151-172, 1990.
[Prior, 1962] A. N. Prior. Formal Logic. Clarendon Press, Oxford, second edition, 1962.
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Approach: Essays in honor of Karl Popper. Free Press of Glencoe, New York, 1963.
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Journal of Symbolic Logic, 33:82-96, 1968.
[Rumfitt, 1997] Ian Rumfitt. The categoricity problem and truth-value gaps. Analysis, 57:223-235, 1997.
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2:149-153, 1961.
[Smiley, 1973] Timothy Smiley. What is a syllogism? Journal of Philosophical Logic, 2:136-154. 1973.
[Smiley, 1994] Timothy Smiley. Aristotle’s completeness proof. Ancient Philosophy, 14:25-38, 1994.
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[Smith, 1982] Robin Smith. What is Aristotelian ecthesis? History and Philosophy of Logic, 3:113-127, 1982.
[Smith, 1989] Robin Smith. Aristotle, Prior Analytics. Hackett, Indianapolis, 1989.
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Liberal Arts, Fukui University, 6:75-87, 1957.
[Sugihara, 1957b] T. Sugihara. Necessity and possiblity in Aristotelian syllogistic. Memoirs of the Faculty of
Liberal Arts, Fukui University, 7:15-22, 1957.
[Thom, 1991] Paul Thom. The two Barbaras. History and Philosophy of Logic, 12:135-149, 1991.
[Thom, 1993] Paul Thom. Apodeictic ecthesis. Notre Dame Journal of Formal Logic, 34:193-208, 1993.
[Thom, 1994] Paul Thom. Interpreting Aristotle's contingency-syllogistic. In C. C. W. Taylor, editor, Oxford
Studies in Ancient Philosophy, volume 12, pages 91-109. Clarendon Press, Oxford, 1994.
[Thom, 1996] Paul Thom. The Logic of Essentialism: An Interpretation of Aristotle's Modal Syllogistic, vol¬
ume 43 of The New Synthese Historical Library. Kluwer, Dordrecht, 1996.
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Logic, 26:129-141, 1997.
[van Rijen, 1989] Jeroen van Rijen. Aspects of Aristotle’s Logic of Modalities. Reidel, Dordrecht, 1989.
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INDIAN LOGIC
Jonardon Ganeri
1 ARGUMENTATION WITHIN DIALECTIC AND DEBATE: PRAGMATIC
CRITERIA FOR GOOD ARGUMENTATION
1.1 Early dialogues: information-seeking, interrogation and cross¬
checking
The intellectual climate of ancient India was vibrant, and bristled with contro¬
versy. Debates were held on a great variety of matters, philosophical, scientific
and theological. Quite soon, the debates became formal affairs, with reputations
at stake and matters of importance in the balance. Already in the Brhadaranyaka
Upanisad (c. 1 th century BCE), we find the sage Yajnavalkya being quizzed by
the king’s priestly entourage on tricky theological puzzles:
Once when Janaka, the king of Videha, was formally seated, Yajnvalkya
came up to him. Janaka asked him: ‘Yajnvalkya, why have you come?
Are you after cows, or discussion about subtle truths?’ He replied:
l Both, your majesty.’ (BU 4.1.1).
What followed was a question-answer type dialogue in which Janaka interro¬
gated the sage, not only to solicit information but to test Yajnavalkya’s mettle.
The sage had earlier granted Janaka a wish, and the wish he chose was the freedom
to ask any question at will. Yajnavalkya was not to be released from this wish
until he had fully satisfied Janaka’s probing inquiry:
[Janaka] ‘Here, sire, I’ll give you a thousand cows! But you’ll have
to tell me more than that to get yourself released!’ At this point
Yajnvalkya became alarmed, thinking: ‘The king is really sharp! He
has flushed me out of every cover.’ (BU 4.3.33-4).
It is in fact a characteristic of the earliest recorded debates that they take the
form of question-answer dialogues. As a form of debate, the goal of a question-
answer dialogue is not restricted merely to one party soliciting information from
another, for there are, as this dialogue shows, elements too of testing out one’s op¬
ponent and cross-checking what he says. A particularly important early question-
answer dialogue is the Milinda-panha , or Questions of King Milinda. It records
the encounter between a Buddhist monk Nagasena and Milinda, also known as
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
@ 2004 Elsevier BV. All rights reserved.
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Menander, an Indo-Bactrian king who ruled in the part of India that had fallen
under Greek influence at the time of Alexander’s Indian campaign. The document
dates from around the first century CE, although Milinda’s reign was 155-130
BCE. At the outset, Nagasena insists that their dialogue is conducted as scholarly
debate and not merely by royal declaration 1 —
King Milinda said: Reverend Sir, will you discuss with me again?
Nagasena: If your Majesty will discuss ( vada) as a scholar, well, but if
you will discuss as a king, no.
Milinda: How is it then that scholars discuss?
Nagasena: When scholars talk a matter over one with another, then is
there a winding up, an unravelling, one or other is convicted
of error, and he then acknowledges his mistake; distinctions
are drawn, and contra-distinctions; and yet thereby they are
not angered. Thus do scholars, 0 King, discuss.
Milinda: And how do kings discuss?
Nagasena: When a king, your Magesty, discusses a matter, and he
advances a point, if any one differ from him on that point, he
is apt to fine him, saying “Inflict such and such a punishment
upon that fellow!” Thus, your Magesty, do kings discuss.
Milinda: Very well. It is as a scholar, not as a king, that I will discuss.
(MP 2.1.3).
Vada, the type of dialogue Nagasena depicts as that of the scholar, is one in
which there are two parties. Each defends a position with regard to the matter
in hand; there is an ‘unravelling’ ( nibbethanam t; an unwinding, an explanation)
and a disambiguation of the positions of both — a process of revealing commit¬
ments, presumptions and faulty argument; there is also a ‘winding up’ ending in
the censure ( niggaho ; Skt. nigraha) of one party, a censure based on reasons he
himself will acknowledge ( patikamman ; ‘re-action’, rejoinder). This is a species of
the persuasion dialogue, a ‘conversational exchange where one party is trying to
persuade the other part that some particular proposition is true, using arguments
that show or prove to the respondent that the thesis is true’ 2 . Indeed, it would
seem to be the species that has come to be known as the critical discussion, a
’A similar distinction, in the types of scientific debate held between physicians, will be drawn
a little later by Caraka, a medical theorist, and an important source of information about ancient
Indian logic. He says, in an echo of the Meno 7.5 c-d, that debate ( sambhasa ) among specialists
is of two types — friendly ( sandhaya ) and hostile ( vigrhya ). See Caraka-Samhita 3.8.16-17
and Ernst Prets, ‘Theories of Debate, Proof and Counter-Proof in the Early Indian Dialectical
Tradition’, in Balcerowicz, Piotr & Mejor, Marek eds., On the Understanding of other cultures:
Proceedings of the International Conference on Sanskrit and Related Studies to Commemorate
the Centenary of the Birth of Stanislaw Schayer, Warsaw 1999. (Warsaw: Oriental Institute,
Warsaw University, 2000).
2 Douglas Walton, The New Dialectic: Conversational Contexts of Argument (Toronto: Uni¬
versity of Toronto Press, 1998), p. 37.
Indian Logic
311
persuasion dialogue in which the conflict is resolved ‘only if somebody retracts his
doubt because he has been convinced by the other party’s argumentation or if he
withdraws his standpoint because he has realized that his argumentation cannot
stand up to the other party’s criticism’ 3 . Not every persuasion dialogue need end
in one party recognising defeat, for an important function of the general persua¬
sion dialogue is to be maieutic, helping each side to clarify the nature of their
commitments and the presuppositions upon which their positions depend. 4 In the
to-and-fro of such a dialogue, each party is allowed to retract earlier commitments,
as it becomes clear what the consequences of such a commitment would be. This
maieutic, clarificatory function of a dialogue is perhaps what Nagasena intends
when he speaks of an ‘unravelling’, and it seems clearer still in his characterisation
of ‘investigation’ ( vikara ) as a ‘threshing-out’:
Milinda: What is the distinguishing characteristic, Nagasena, of reflec¬
tion ( vitakka )?
Nagasena: The effecting of an aim.
Milinda: Give me an illustration.
Nagasena: It is like the case of a carpenter, great king, who fixes in
a joint a well-fashioned piece of wood. Thus it is that the
effecting of an aim is the mark of reflection.
Milinda: What is the distinguishing characteristic, Nagasena, of inves¬
tigation (vikara)?
Nagasena: Threshing out again and again.
Milinda: Give me an illustration.
Nagasena: It is like the case of the copper vessel, which, when it is
beaten into shape, makes a sound again and again as it grad¬
ually gathers shape. The beating into shape is to be regarded
as reflection and the sounding again and again as investiga¬
tion. Thus it is, great king, that threshing out again and
again is the mark of investigation.
Milinda: Very good, Nagasena. (MP 2.3.13-14).
So it is through reflection and argumentation that the parties to an investiga¬
tion together thrash out a position. Nagasena tells us very little about the sort
of argumentation that is appropriate, and we can learn little more about argu¬
ment within persuasion dialogues from the Questions of King Milinda (although
Milinda’s repeated request to be given an illustration is suggestive of the impor¬
tance that would later be attached to the citation of illustrative examples in good
argumentation; see §1.3 below). And yet there is still something to learn. For
3 Frans van Eemeren & Rob Grootendorst, Argumentation, Communication and Fallacies
(Hillsdale: Lawrence Erlbaum Associates,1992), p. 34.
4 Walton (1988: 48).
312
Jonardon Ganeri
the dialogue of the Questions of King Milinda is not, contrary to Nagasena’s ini¬
tial statement, a straighforwardly scholarly debate, but proceeds instead with his
being interrogated at the hands of Milinda. Ostensibly Milinda wishes to be in¬
formed as to the answer to a range of thorny ethical and metaphysical questions,
but his questioning is not so innocent, and at times he seems intent on entrapping
Nagasena in false dichotomies and leading questions. So it is said of him:
Master of words and sophistry ( vetandi ), clever and wise
Milinda tried to test great Nagasena’s skill.
Leaving him not, again and yet again,
He questioned and cross-questioned him, until
His own skill was proved foolishness. (MP 4.1.1).
Milinda here is significantly described as a ‘master of sophistry’ or vetandi, a
practitioner of the dialogue form known as vitanda, a ‘refutation-only’ type of
dialogue in which the opponent defends no thesis of his own but is set only on re¬
futing that of the proponent (see §1.4). The implication here is that such dialogues
are essentially eristic. And it is, in particular, the eristic use of questioning that
Milinda is a master of. Questions need not be innocent requests for information;
they can also be disguised arguments. To reply to the question ‘When did you stop
cheating on your tax returns?’ at all, affirmatively or negatively, is already to com¬
mit oneself to the ‘premise’ of the question, that one has indeed been cheating on
one’s tax returns. In the intellectual climate of ancient India, when interrogative
dialogue was common-place, it was very well known that questions can be used to
entrap the unwitting, and counter-strategies were invented to avoid entrapment.
The Buddha himself was well aware that replying to a yes-no question can commit
one to a proposition, whatever answer one gives, and his solution, famously, was
to refuse to answer. Thus when asked a series of ten leading questions — is the
soul is eternal? is it non-eternal? etc. — the Buddha declined to offer a reply. For
any reply would commit him, against his wish, to the existence of souls. In the
Questions of King Milinda, we see Nagasena experimenting with a different tech¬
nique to avoid entrapment. To some of Milinda’s more devious yes-no questions,
instead of refusing to reply at all, Nagasena replies ‘Both yes and no’! To others
he replies ‘Neither yes nor no’! For example:
Milinda: He who is born, Nagasena, does he remain the same or be¬
come another?
Nagasena: Neither the same nor another.
Milinda: Give me an illustration.
Nagasena: Now what do you think, O king? You were once a baby, a
tender thing, and small in size, lying flat on your back. Was
that the same as you who are now grown up?
Milinda: No. That child was one, I am another.
Indian Logic
313
Nagasena: If you are not that child, it will follow that you have had
neither mother nor father, no! nor teacher. You cannot
have been taught either learning, or behaviour, or wisdom.
... Suppose a man, 0 king, were to light a lamp, would it
burn the night through?
Milinda: Yes, it might do so.
Nagasena: Now, is it the same flame that burns in the first watch of
the night, Sir, and in the second?
Milinda: No.
Nagasena: Or the same that burns in the second watch and the third?
Milinda: No.
Nagasena: Then there is one lamp in the first watch, and another in
the second, and another in the third?
Milinda: No. The light comes from the same lamp all the night through.
Nagasena: Just so, O king, is the continuity of a person or thing main¬
tained. One comes into being, another passes away; and the
rebirth is, as it were, simultaneous. Thus neither as the same
nor as another does a man go on to the last phase of his
self-consciousness. (MP 2.2.1)
The ‘premise’ of the question, that to change is to cease to be, is very effectively
refuted with a ‘neither yes nor no’ reply. Nagasena first makes Milinda acknowledge
that, with this as the background premise, answering either ‘yes’ or ‘no’ leads to an
absurdity. For if he is strictly identical to the child, then he must share that child’s
properties; and if he is different, then he cannot. Having exposed the false premise,
Nagasena, rejects it in favour of the view that persistence through time requires
not strict identity but causal continuity. Here is a different kind of example:
Milinda: Does memory, Nagasena, always arise subjectively, or is it
stirred up by suggestion from outside?
Nagasena: Both the one and the other.
Milinda: But does not that amount to all memory being subjective in
origin, and never artificial?
Nagasena: If, 0 king, there were no artificial (imparted) memory, then
artisans would have no need of practice, or art, or schooling,
and teachers would be useless. But the contrary is the case.
Milinda: Very good, Nagasena. (MP 3.6.11).
Here the question’s hidden premise is that memories are caused either wholly
by what goes on in the mind or wholly by factors external to it, and the ‘both
yes and no’ reply makes plain that what ought to be said is that memories are
314
Jonardon Ganeri
wholly caused either by what goes on in the mind or by factors external to it, but
not caused wholly by one or the other. Again, subsidiary argumentation exposes
the absurdity in replying with an unqualified ‘yes’ or an unqualified ‘no’. It was
perhaps in recognition of the tactical importance of such ‘neither yes nor no’ and
‘both yest and no’ replies that it became a common-place that there are four
possible ways of responding to any question of the yes-no type, an idea that was
systematised in the work of Nagarjuna (§1-4). What we see very clearly in the
Questions of King Milinda is a sophisticated early appreciation of the pragmatics
of interogative dialogues.
1.2 On balance and fairness in the conduct of dialogue:
The Kathavatthu
The Kathavatthu or Points of Controversy (circa third century BCE) is a book
about method. It describes, for the benefit of adherents to various Buddhist
schisms, the proper method to be followed in conducting a critical discussion into
an issue of doctrinal conflict. Recent scholarship has largely focussed on the ques¬
tion of the extent to which there is, in the Kathavatthu, an ‘anticipation’ of results
in propositional logic. 5 For, while it is true that the formulation of arguments
there is term logic rather than propositional, and true also that the propositional
rules are nowhere formulated in the abstract, the codified argumentation clearly
exploits manipulations that trade on the definition of material implication, on con¬
traposition, and on at least one of modus tollens, modus ponens and reductio ad
absurdum. The preoccupation with this question of anticipation, assumes, how¬
ever, a methodology for the interpretation of Indian logic that suffers a number of
serious disadvantages. For, first, in presupposing that the only matter of interest
is the extent to which a given text displays recognition of principles of formal logic,
the methodology fails to ask what it was that the authors themselves were trying
to do, and in consequence, is closed to the possibility that these texts contribute
to logical studies of a different kind. And second, in supposing that arguments
have to be evaluated formally, the important idea that there are informal criteria
for argument evaluation is neglected. In fact, the Kathavatthu offers a particularly
clear example of a text whose richness and interest lies elsewhere than in its antic¬
ipation of deductive principles and propositional laws. As a meticulous analysis of
the argumentation properly to be used in the course of a dialogue of a specific type,
its concern is with the pragmatic account of argument evaluation, the idea that
5 Aung, S.Z., Points of Controversy, or, Subjects of Discourse: Being a Translation of the
Kathavatthu from the Abhidhammapitaka, eds. S.Z. Aung and C.A.F. Rhys Davids. Pali Text So¬
ciety, translation series no.5. London: Luzac &: Co. 1915; reprint 1960; Schayer, St., “Altindische
Antizipationen der Aussagenlogik”, Bulletin international de I’Academie Polonaise des Sciences
et des Lettres, classe de philologies: 90-96 (1933), translated in Jonardon Ganeri ed., Indian
Logic: A Reader (London: Curzon, 2001); Bochenski, J. M., “The Indian Variety of Logic”, in
his A History of Formal Logic. Freiburg. Trans. 1. Thomas, Notre Dame: University of Notre
Dame Press (1961), pp. 416-447., reprinted in Jonardon Ganeri ed., Indian Logic: A Reader,
Matilal, Bimal Krishna, The Character of Logic in India. Albany: State University of New York
Press, 1998.
Indian Logic
315
arguments have to be evaluated as good or bad with regard to their contribution
towards the goals of the dialogue within which they are embedded. The leading
concern of the Kathavatthu is with issues of balance and fairness in the conduct
of a dialogue, and it recommends a strategy of argumentation which guarrantees
that both parties to a point of controversy have their arguments properly weighed
and considered. It is important, in the normative framework of the Kathavatthu ,
that there is a distinction between the global aim of the dialogue as a whole —
here to rehearse in an even-handed manner all the considerations that bear upon
an issue of dispute, to clarify what is at stake even if no final resolution is achieved
— and the local aim of each participant — to advocate the stance they adopt with
regard to that issue by supplying arguments for it and attacking the arguments of
the other parties.
A dialogue conducted in accordance with the prescribed method of the
Kathavatthu is called a vadayutti. The goal of a vadayutti is the reasoned ex¬
amination (yuttr, Skt. yukti) of a controversial point in and through a noneristic
dialogue ( vada ). The dialogue is highly structured, and is to be conducted in ac¬
cordance with a prescribed format of argumentation. There is a given point at
issue, for example, whether ‘a person is known in the sense of a real and ulti¬
mate fact’ (i.e. whether persons are conceived of as metaphysically irreducible),
whether there are such things as morally good and bad actions, and so, in general,
whether A is B. A dialogue is now divided into eight sub-dialogies or ‘openings’
(atthamukha). These correspond to eight attitudes it is possible to adopt with
regard to the point at issue. So we have:
[1] Is A B?
[2] Is A not B?
[3] Is A B everywhere?
[4] Is A B always?
[5] Is A B in everything?
[6] Is A not B everywhere?
[7] Is A not B always?
[8] Is A not B in everything?
The introduction of an explicit quantification over times, places and objects
serves to determine an attitude of proponent and respondent to the point of con¬
troversy. If the issue in question is, for example, whether lying is immoral, the
clarification would be as to whether that proposition is to be maintained or de¬
nied, and in either case, whether absolutely, or only as relativised in some way to
circumstances, times or agents. So an opening thesis here is by definition a point
at issue together with an attitude towards it.
316
Jonardon Ganeri
Each such‘opening’ now proceeds as an independent dialogue, and each is di¬
vided into five stages: the way forward ( anuloma ), the way back ( patikamma ),
the refutation ( niggaha ), the application ( upanayana ), and the conclusion ( nigga-
mana). In the way forward, the proponent solicits from the respondent their
endorsement of a thesis, and then tries to argue against it. In the way back, the
respondent turns the tables, soliciting from the proponent their endorsement of
the counter-thesis, and then trying argue against it. In the refutation, the respon¬
dent, continuing, seeks to refutes the argument that the proponent had advanced
against the thesis. The application and conclusion repeat and reaffirm that the
proponent’s argument against the respondent’s thesis is unsound, while the re¬
spondent’s argument against the proponent’s counter-thesis is sound.
It is significant to note that there is here no pro-argumentation, either by the
respondent for the thesis or by the proponent for the counter-thesis. There is only
contra-argumentation, and that in two varieties. The respondent, in the ‘way back’,
supplies an argument against the proponent’s counter-thesis, and in the refutation
stage, against the proponent’s alleged argument against the thesis. So we see here
a sharp distinction between three types of argumentation - pro argumention, ar¬
gumentation that adduces reasons in support of one’s thesis, counter argumenation
— argumentation that adduces reasons against the counter-thesis, and defensive
argumentation, argumentation that defends against counter-arguments directed
against one’s thesis. The respondent, having been ‘attacked’ in the first phase,
‘counter-attacks’ in the second phase, ‘defends’ against the initial attack in the
third, and ‘consolidates’ the counter-attack and the defence in the fourth and
fifth. The whole pattern of argumentation, it would seem, is best thought of as
presumptive, that is, as an attempt to switch a burden of proof that is initially
even distributed between the two parties. The respondent tries to put the bur¬
den of proof firmly onto the proponent, by arguing against the proponent while
countering any argument against herself. The fact that the respondent does not
offer any pro argumentation in direct support of the thesis means that the whole
pattern of argumentation is technically ab ignorantium-, that is, argumentation of
the form “I am right because not proved wrong”. But ab ignorantium reasoning
is not always fallacious; indeed, it is often of critical importance in swinging the
argument in one’s favour in the course of a dialogue (see §1-5).
In the first stage, the way forward, the proponent elicits from the respondent
an endorsement of a thesis, and then sets out to reason against it. Not any form
of reasoning is allowed; indeed the Kathavatthu prescribes a very specific method
of counter-argumentation. Thus:
I. The Way Forward
Theravadin: Is the soul ( puggala ) known as a real and ultimate fact?
[1] Puggalavadin : Yes.
Theravadin: Is the soul known in the same way as a real and ultimate fact is
known?
[2] Puggalavadin: No, that cannot be truly said.
Indian Logic
317
Theravadin: Acknowledge your refutation ( niggaha ):
[3] If the soul be known as a real and ultimate fact, then indeed, good sir, you
should also say, the soul is known in the same way as any other real and ultimate
is known.
[4] That which you say here is false, namely, that we should say, “the soul is known
as a real and ultimate fact”, but we should not say, “the soul is known in the same
way as any other real and ultimate fact is known.”
[5] If the later statement cannot be admitted, then indeed the former statement
should not be admitted either.
[6] In affirming the former, while denying the latter, you are wrong.
The respondent, here a puggalavadin or believer in the existence of personal
souls, is asked to endorse the thesis. The proponent then attempts to draw out an
implication of that thesis, an implication more over to which the puggalavadin will
not be willing to give his consent. Here the thesis that persons are thought of as
metaphysically irreducible elements of the world is held to imply that knowledge
of persons is knowledge of the same kind as that of other types of thing. The
puggalavadin , will perhaps want to draw an epistemological distinction between
empirical knowledge of external objects and self-knowledge, and so will not endorse
this derived proposition. And now the proponent, in a fresh wave of argumenta¬
tion, demonstrates that it is inconsistent for the puggalavadin to endorse the thesis
but not the derived consequence. So a counter-argument has three components:
the initial thesis or thapana (Skt. sthdpana ), the derived implication or papana,
and the demonstration of inconsistency or ropana.
It is in the ropana that there seems to be an ‘anticipation’ of propositional logic.
Of the four steps of the ropana, the first, from [3] to [4], looks like an application
of the definition of material implication or its term-logical equivalent:
(A is B — > ( C is D) =defn ~■((A is is D)).
Notice here that an effect of soliciting from the respondent a ‘no’ in answer to
the proponent’s second question is that the negation is external and not internal.
Thus, we have ‘~’(C is D)' rather than ‘((7 is -> D)\ This what one needs in the
correct definition of material implication.
The second step, from [4] to [5], looks like a derivation of the contraposed
version of the conditional, a derivation that depends on the stated definition of
the conditional. From that definition, and assuming that is commutative, it
follows that
(. A is B) —► ( C is D) iff ->((7 is D ) —> -i(A is B).
The final step now is an application of modus ponens. So what we have is:
[1] {A is B) premise
[2] -'((7 is D ) premise
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[3] (A is B) -4 (C is D)
additional premise?
[4] -,((^ is B)k->{C is D))
[5] -i(C is D) -4 ->(A is B)
[6] -i(j 4 is B)
3, defn. of -4
4, defn. of -4
2, 5, MP
This is how Matilal 6 reconstructs the ropana stage of argumentation. Earlier,
Bochenski 7 recommended a variant in which steps [3] and [4] “together constitute
a kind of law of contraposition or rather a modus tollendo tollens in a term-logical
version”. Still another alternative is to see step [3] as a piece of enthymematic
reasoning from the premise already given, rather than as the introduction of an
additional premise. In other words, the ‘if. ..then’ in [3] is to be understood to
signify the logical consequence relation rather than material implication. Then
step [4] negates the premise in an application of reductio ad absurdum. That is:
[1,2] {A is B) k -.(C is D)
[3] (C is D)
[4] -.((A is B) k -.(C is D))
[5] -i (C is D) -4 -(A is B)
[6] -.((A is B) k -(C is D))
premise
1 + 2, enthymematic derivation
1 + 2,3; RAA
4, defn. of -4
5, defn. of -4
This reconstuction seems more in keeping with the overall pattern of argumen¬
tation — to take the respondent’s thesis and derive from it consequences the re¬
spondent will not endorse, and thereby to argue against the thesis (and it preserves
the repetition of the original). Here again we see that the form of argumentation in
the Kathdvatthu is better understood if we bear in mind the function it is intended
to serve within a dialogue context.
The same dialogue context is normative, in the sense that it gives the grounds
for evaluating any actual instance of such argumentation as good or bad. It seems
possible to understand the ‘way forward’ in terms of certain concepts from the
theory of argumenation. Hamblin introduced the idea that each participant in a
dialogue has a ‘commitment store’, a set of propositions to which they commit
themselves in the course of the dialogue, primarily by asserting them directly. 8
In Hamblin’s model, the commitments of each party are on public display, known
to every participant in the dialogue. In order to represent the fact that this is
very often not the case, Walton 9 employs a distinction between open or ‘light-side’
commitments, and veiled or ‘dark-side’ commitments. The veiled commitments of
6 Matilal (1998: 33-37)
7 Bochenski (1961: 423)
®Hamblin, C. L., Fallacies. London: Methuen, 1970.
9 Walton (1998: 50-51).
Indian Logic
319
a participant are not on public view, and might not be known even to that partic¬
ipant themselves: but perhaps the participant trades on them in making certain
kinds of dialogue move. Indeed, it is part of what Walton 10 calls the ‘maieutic’
role of dialogue to make explicit the veiled commitments of the participants, a
process of clarification that is valuable even if it does not lead to the issue at stake
being decided in favour of one party or the other. 11
Something of this sort is what is being described in the initial stages of the
‘way forward’. Steps [1] and [2] elicit from the respondent an explicit and open
commitment to the propositions ‘A is B ’ and (C is D)’. ^,From the respective
assertion and denial, these become parts of her explicit commitment store. But
next, though the enthymematic argumentation that constitutes the papana or
stage [3], it is made clear that the respondent has a veiled commitment to the
proposition ‘C is D’. For this is shown to follow from propositions in the explicit
commitment store of the respondent. Finally, the ropana stage of reasoning reveals
this newly explosed commitment to be inconsistent with the respondent’s other
explicit commitments. The overall effect is to force the respondent into a position
where she must retract at least one of the propositions to which she has committed
herself. Indeed, we can say that such a retraction is the primary goal of the way
forward. The primary aim is not to disprove the thesis, but to force a retraction
of commitment. So when we evaluate the argumentation used in this part of
the dialogue, it is to be evaluated as good or bad with reference to how well it
succeeds in forcing such a retraction, and not simply or only or even in terms of its
deductive or inductive soundness. The strategic problem here is how to persuade
the respondent to accept some proposition that is meant ultimately to be used to
force a retraction, and the type of strategy being recommended is the one Walton
calls that of “separating”, where “two or more propositions are proved separately
and then eventually put together in an argument structure that is used to prove
one’s own thesis or argue against an opponent’s”. 12 In setting out the reasoning
in this way, the intention of the author of the Kathavatthu is not to imply that
precisely this sequence of arguments is sound. What is being shown is the form
that any counter-argument should take. It is a description, in generic terms, of
the strategic resources open to the proponent, and serves rather as a blue-print
for any actual vadayutti dialogue.
At this point in the sub-dialogue that is the first opening, then, the burden of
proof seems to lie squarely with the respondent, the puggalavadin, who is being
pressured into the uncomfortable position of having to retract his stated thesis.
The remaining four phases of the first opening are a summary of the strategic
resources open to the respondent to recover his position, and indeed to turn the
tables against the proponent. First, the way back. This is a phase of counter-
10 Walton (1998: 58).
11 The term ‘maieutic’, from maieutikos ‘skill in midwifery, is taken from the Theaetetus , where
Socrates describes himself as a midwife for beautiful boys - helping them to give birth to whatever
ideas are in them, and test them for whether they are sound.
12 Walton (1998: 44).
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attack, in which the respondent uses parallel reasoning to force the proponent too
into a position of retraction with regard to the counter-thesis.
II. The Way Back
Puggalavadin: Is the soul not known as a real and ultimate fact?
[1] Theravadin: No, it is not known.
Puggalavadin: Is it not known in the same way as any real and ultimate fact is
known?
[2] Theravadin: No, that cannot be truly said.
Puggalavadin: Acknowledge the rejoinder ( patikamma ):
[3] If the soul is not known as a real and ultimate fact, then indeed, good sir, you
should also say: it is not known in the same way as any other real and ultimate
fact is known.
[4] That which you say is false, namely, that we should say “the soul is not known
as a real and ultimate fact”, but we should not say “it is not known in the same
way as any other real and ultimate fact is known”.
[5] If the latter statement cannot be admitted, then indeed the former statement
should not be admitted either.
[6] In affirming the former while denying the latter, you are wrong.
At the end of the ‘way back’, if the respondent’s arguments have gone well, the
proponent has been pressed in the direction of retracting his commitment to the
counter-thesis. If the respondent were to leave matters here, however, he would
have failed in the global aim of the ‘opening’. The aim of the opening is to shift
the burden of proof decisively onto the proponent. After the second stage in the
opening, however, the burden of proof is again symmetrically distributed among
the parties to the dialogue — both are in a position of being pressed to retract
their respective commitment. So, in the third phase, the respondent seeks, in
a defensive move, to diffuse the argument of the proponent that is forcing this
retraction. Again, the cited reasoning is schematic, it indicates a general strategy
the details of which must be worked out differently in each specific case. The
distinction being drawn is the one between counter-argument, and defensive repost ,
a distinction that makes sense only within the normative framework of a dialogical
exchange.
The first opening in the vadayutti has rehearsed the best argumentation that
available against someone whose attitude towards the point at issue is one of
unqualified affirmation. Remember, however the global aim of a vadayutti — to
be the form of dialogue most conducive to a balanced examination of the best
arguments, both for and against. It is the function now of the second opening
to rehearse the best argumentation against someone whose attitude towards the
point at issue is one of unqualified denial, and of the subsequent openings to do
likewise with respect to attitudes of qualified affirmation and denial. Even at the
end of the dialogue, there may be no final resolution, but an important maieutic
function has been served — the clarification of the commitments entailed by each
position, of their best strategies and forms of argumentation. So, indeed, it is
Indian Logic
321
as a rich account of presumptive reasoning in dialogue, and not so much for its
‘anticipations’ of formal logic, that the Kathavatthu makes a rewarding object of
study.
1.3 Case-based, reasoning, extrapolation and inference from sampling:
The Nyayasutra
It was Henry Colebrooke 13 who first brought Indian logic to the attention of the
English philosophical world, announcing in a famous lecture to the Royal Asiatic
Society in 1824 his discovery of what he called the ‘Hindu Syllogism’. Colebrooke’s
‘discovery’ consisted in fact in a translation of an ancient Indian treatise called
the Nyayasutra. It dates from around the 1 st or 2 nd century AD, and is said to
be the work of Gautama Aksapada. Scholars are now inclined to regard it as the
amalgamation of two earlier works on philosophical method, one a treatise on the
rules and principles of debate, the other a discussion of more general issues in
epistemology and metaphysics. In a section on the proper way for a debater to set
out their argument, the Nyayasutra prescribes a five-step schema for well-formed
argument, and it is this schema that Colebrooke identified as the Indian syllogism.
We now know much much more than Colebrooke about the historical development
of Indian logic. He, for instance, was unaware of the informal logic and anticipa¬
tions of propositional calculus in the Kathavatthu (§1.2), or the theories of the
Buddhists Vasubandhu, Diiinaga and Dharmaklrti on formal criteria for inference
(§§2.1-5). And scholars had yet to learn the complexities of the later logical school
of Navya-Nyaya (§§4.1-3), with its intriguing treatment of negation, logical conse¬
quence and quantification, and even, as Daniel Ingalls has shown in his pioneering
book entitled Materials for the Study of Navya-Nyaya Logic, the formulation of
De Morgan’s Laws. 14 Nevertheless, in spite of Colebrooke’s lack of acquaintance
with the historical context, he and those who followed him were right to see the
Nyayasutra as a treatise of fundamental importance in Indian logical thinking,
and I would like to pick up and continue the thread of their discussion. I want to
argue that the Nyayasutra begins a transformation in Indian thinking about logic.
And this in two inter-related respects: in the beginnings of a shift of interest away
from the place of argumentation within dialectic and debate and towards a greater
concern with the more formal properties of sound inference ; and in a parallel and
correlated shift from case-based to rule-governed accounts of logical reasoning. I
will discuss each of these in turn.
In the Nyayasutra, there is a more systematic discussion of the categories and
methods of debate than in earlier debating manuals. Three kinds of debate are
13 H. T. Colebrooke, “On the Philosophy of the Hindus: Part II - On the Nyaya and Vaiseshika
systems”. Transactions of the Royal Asiatic Society (1824), 1: 92-118; reprinted in Jonardon
Ganeri ed., Indian Logic: A Reader.
14 D. H. H. Ingalls, Materials for the Study of Navya-Nyaya Logic (Cambridge Mass.: Harvard
University Press), 1951, pp. 65-67.
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distinguished: good or honest debate ( vada ), tricky or bad debate ( jalpa ) and a
refutation-only debate ( vitanda ):
Good debate (vada) is one in which there is proof and refutation of the¬
sis and antithesis based on proper evidence ( pramana) and presumptive
argumentation ( tarka ), employing the five-step schema of argumenta¬
tion, and without contradicting any background or assumed knowledge
( siddhanta ).
Tricky debate (jalpa) is one in which, among the features mentioned
before, proof and refutation exploit such means as quibbling ( chala ),
false rejoinders (jdti), and any kind of clincher or defeat situation (ni-
grahasthana).
Refutation-only debate (vitanda) is one in which no counter-thesis is
proven. (NS 1.2.1-3).
Here is our first reference to the Indian five-step inference pattern. It is a schema
for proper argumentation among disputants who are engaged in an honest, friendly,
noneristic, and balanced debate (vada). In the dialectical context in which such
arguments are embedded, a proponent attempts to prove a thesis and to refute
the counter-thesis of the opponent, both parties drawing-upon a shared body of
background knowledge and received belief, and using properly accredited methods
for the acquisition and consideration of evidence. The aim of each participant
in the dialogue is not victory but a fair assessment of the best arguments for
and against the thesis. In Indian logic, vada represents an ideal of fair-minded
and respectful discourse. By contrast, in a tricky debate (jalpa), underhanded
debating tactics are allowed, and the aim is to win at all costs and by any means
necessary. The third kind of debate, the refutation-only debate (vitanda), is the
variety of dialogue preferred by the sceptics — to argue against a thesis without
commitment to any counter-thesis. It is not entirely clear whether this is a type of
good or tricky debate. We might conjecture, however, that if dialectic is a rough
kin of vada, and sophistic of jalpa, then the Socratic elenchus could be regarded
as a species of vitanda, which is not, therefore, an entirely disreputable method of
debate.
The aim, in a good debate between friends, is the assessment of the best argu¬
ments for or against the thesis. And that leads to the question: how are arguments
to be assessed or evaluated? Are the criteria for assessment formal, to do only with
the form of the argument schema itself; or are they informal, pragmatic criteria,
according to which arguments have to be evaluated as good or bad with regard to
their contribution towards the goals of the dialogue within which they are embed¬
ded?
With this question in mind, let us look at the five-step proof pattern. The proper
formulation of an argument is said to be in five parts: tentative statement of the
thesis to be proved (pratijna)-, citation of a reason (hetu)- mention of an example
(udaharana ); application of reason and example to the case in hand (upanaya)\
Indian Logic
323
final assertion of the thesis (nigamana). An unseen fire is inferred to be present on
the mountain, on the basis of a plume of smoke; just as the two have been found
associated in other places like the kitchen. The terms used here are defined in a
series of admittedly rather gnomic utterances (NS 1.1.34-39):
1.1.32 ‘the parts [of an argument scheme] are thesis, reason, example, application
and conclusion’
( pratijndhetuddharanopanayanigamandnyavayavah ).
1.1.33 ‘the thesis is a statement of that which is to be proved’
(sadhyanirdesahpratijha).
1.1.34 ‘the reason is that which proves what is to be proven in virtue of a simi¬
larity with the example’ ( udaharanasadharmyat sadhyasadhanam hetuh).
1.1.35 ‘again, in virtue of a dissimilarity’ ( tathd vaidharmyat).
1.1.36 ‘the example is an illustration which, being similar to that which is to
be proved, has its character’ ( sadhyasadharmyat taddharmabhavi drstanta
udaharanam).
1.1.37 ‘or else, being opposite to it, is contrary’ ( tadviparyayad va vipantam).
1.1.38 ‘the application to that which is to be proved is a drawing in together
(upasamhara ) “this is so” or “this is not so,” depending on the example’
(udaharanapeksas tathety upasamha.ro na tatheti va sadhyasyopanayah).
1.1.39 ‘the conclusion is a restatement of the thesis as following from the state¬
ment of the reason’ ( hetvapadesat pratijhdyah punarvacanam nigamanam).
The basic idea is that an object is inferred to have one (unobserved) property
on the grounds that it has another, observed, one — “there is fire on the mountain
because there is smoke there”. The most distinctive aspect of the schema, though,
is the fundamental importance given to the citation of an example, a single case
said either to be similar or else dissimilar to the case in hand. Suppose I want
to persuade you that it is about to rain. I might reason as follows: “Look, it is
going to rain (thesis). For see that large black cloud (reason). Last time you saw
a large black cloud like that one (example), what happened? Well, its the same
now (application). It is definitely going to rain (conclusion).”
Let us try to unpick the Nyayasutra definitions. Suppose we let ‘F’ denote the
property that serves as the reason here ( hetv .), ‘G’ the property whose presence
we are seeking to infer ( sadhya ), ‘a’ the new object about which we are trying to
decide if it is G or not ( paksa ), and ‘6’ the cited example ( udaharana ). Then we
seem to have a pair of schematic inferences, one based on similarity, the other on
dissimilarity:
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Five-step proof based on similarity
[thesis] Ga
[reason] Fa proves Ga, because b is similar to a.
[example] b has the ‘character of a ’ because it is similar to a.
[application] a is the same as b with respect to G.
[conclusion] Ga
Five-step proof based on dissimilarity
[thesis] Ga
[reason] Fa proves Ga, because b is dissimilar to a.
[example] b does not have the ‘character of a’ because it is dissimilar to a.
[application] a is not the same as b with respect to G.
[conclusion] Ga
The counter-proof follows the same pattern, proving the counter-thesis (-> Ga)
by means of a different reason and example:
Counter-proof
[thesis] ~'Ga
[reason] F’a proves Ga, because b is similar to a.
[example] c has the ‘character of a’ because it is similar to a.
[application] a is the same as c with respect to G.
[conclusion] -'Ga
The five-step schema was interpreted in a particular way by Vatsyayana, the
first commentator on the Nyayasutra. His interpretation is largely responsible
for shaping the direction Indian logic was later to take. At the same time, it
was an interpretation that made the citation of an example essentially otiose.
Vatsyayana was, in effect, to transform Indian logic, away from what it had been
earlier, namely a theory of inference from case to case on the basis of resemblance,
and into a rule-governed account in which the citation of cases has no significant
role.
Let us then consider first Vatsyayana’s interpretation. What Vatsyayana says
is that the similarity between a and b just consists in their sharing the reason
property F. The basic pattern of inference is now: a is like b [both are F};
Gb Ga. Or else: a is unlike b [one is F and the other isn’t]; ->Gb Ga. Writing
it out as before, what we have is:
Indian Logic
325
Proof based on similarity
[thesis]
Ga
[reason]
Fa
[example]
Fb
[application]
Gb
[conclusion]
Ga
b is similar to a [both are F],
In a counterproof, a is demonstrated to be similar in some other respect to some
other example, one that lacks the property G:
Counterproof
[thesis] ->Ga
[reason] F'a
[example] F'c
[application] ->Gc
[conclusion] -Ga
Thus, for example, a proof might be: the soul is eternal because it is uncreated,
like space. And the counterproof might be: the soul is non-eternal because it is
perceptible, like a pot.
The proposal is that if a resembles b, and b is G, then a can be inferred to
be G too. But there is an obvious difficulty, which is that mere resemblance
is an insufficient ground. Admittedly, the soul and space are both uncreated,
but why should that give us any grounds for transferring the property of being
eternal from one to the other? The respect in which the example and the case
in hand resemble one another must be relevant to the property whose presence is
being inferred. This is where the idea of a ‘false proof’ or ‘false rejoinder’ ( jati )
comes in. Any argument that, while in the form of the five-step schema, fails
this relevance requirement is called a ‘false proof’ and the Nyayasutra has a whole
chapter (chapter 5) classifying and discussing them. A ‘false rejoinder’ is defined
in this way:
NS 1.2.18 ‘a jati is an objection by means of similarity and dissimilarity’
(sadharymavaidharmyabham pratyavasthanam jatih).
It appears to be admissible to transfer the property ‘rain-maker’ from one black
cloud to another black cloud, but not from a black cloud to a white cloud. It
appears to be admissible to transfer the property ‘has a dewlap’ from one cow to
another cow, but not from one four-legged animal (a cow) to another (a horse). It
c is similar to a [both are F'].
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Jonardon Ganeri
is clear what now needs to be said. The argument is good if there exists a general
relationship between the reason F and the property being proved G, such that the
latter never occurs without the former.
It is the Buddhist logician Diiinaga (480-540 CE) who seems to have been the
first to make this explicit (see also §2.2). According to him, a reason must satisfy
three conditions. Define a ‘homologue’ ( sapaksa ) as an object other than a that
possesses G, and a ‘heterologue’ ( vipaksa ) as an object other than a that does not
possess G. Then Dirinaga’s three conditions on a good reason are:
[1] F occurs in a.
[2] F occurs in some homologue.
[3] F occurs in no heterologue.
Condition [3] asserts, in effect, that F never occurs without G, and this, together
with [1] that F occurs in a, implies of course that G occurs in a. In effect, the
citation of an example in the original Nyayasutra formula has been transformed
into a statement of a general relationship between F and G. There remains only
a vestigial role for the example in condition [2], which seems to insist that there
be an instance of F other than a which is also G. Diiinaga is worried about the
soundness of inferences based on a reason which is a property unique to the object
in hand; for example, the inference “sound is eternal because it is audible”. For if
this is sound, then why not the counter-argument “sound is non-eternal because
it is audible”? And yet there are many inferences like this that are sound, so it
seems to be a mistake to exclude them all. In fact condition [2] soon came to be
rephrased in a way that made it logically equivalent to [3], namely as saying that
F occur only in homologues ( eva ‘only’ is used here as a quantifier). In asking for
the respect in which the example and the new case must resemble each other, for
the presence of G in the example to be a reason for inferring that G is present in
the new case, we are led to give the general relationship that any such respect must
bear to G, and that in turn makes citation of an example otiose. The five-step
schema becomes:
[thesis] Ga
[reason] because F
[example] where there is F, there is G; for example, b.
[application] Fa
[conclusion] Ga
It is the five-step argument pattern so transformed that has suggested to Cole-
brooke and other writers on Indian logic a comparison with an Aristotelian syllo¬
gism in the first figure, Barbara. We simply re-write it in this form:
Indian Logic 327
All F are G.
Fa.
Therefore, Ga.
This assimilation seems forced in at least two respects. First, the conclusion of
the Nyayasutra demonstration is a singular proposition. In Aristotle’s system, on
the other hand, it is always either a universal proposition with ‘all’ or ‘no’, or a
particular proposition with ‘some’. Second, and relatedly, the role of the ‘minor
term’ is quite different: in the Indian schema, it indicates a locus for property-
possession, while in Aristotle, the relation is ‘belongs to’. Again, in reducing the
Indian pattern to an Aristotelian syllogism, the role of the example, admittedly
by now rather vestigial, is made to disappear altogether.
A rather better reformulation of the five-step schema is suggested by Stanislaw
Schayer, 15 who wants to see the Indian ‘syllogism’ as really a proof exploiting two
rules of inference:
[thesis]
Ga
There is fire on a (= on this mountain).
[reason]
Fa
There is smoke on a.
[‘example’]
(x)(Fx -a Gx)
For every locus x : if there is smoke in
x then there is fire in x.
[application]
Fa —^ Ga
This rule also applies for x = a.
[conclusion]
Ga
Because the rule applies to x = a and
the statement Ga is true, the statement
Fa is true.
Two inference rules are in play here — a rule of substitution, allowing us to infer
from ‘(x)^x’ to ‘( a ’> and a rule °f separation, allowing us to infer from l 9 —> cp’
and ‘ 6’ to l <p\ Schayer thereby identifies the Indian syllogism with a proof in a
natural deduction system:
THESIS. Ga because Fa.
Proof.
1 (1) Fa Premise
2 (2) ( x)(Fx —* Gx) Premise
3 (3) Fa —» Ga 2, by V Elimination
1,2 (4) Ga 1 & 3, by -» Elimination. ■
We have seen how the Nyayasutra model of good argumentation came to be
transformed and developed by later writers in the Indian tradition in the direc¬
tion of a formal, rule-governed theory of inference, and how writers in the West
have interpreted what they have called the Indian ‘syllogism’. I suggested at the
beginning that we might try to interpret the Nyaya model along different lines
altogether, seeing it an early atttempt at what is now called ‘case-based reason¬
ing’. Case-based reasoning begins with one or more prototypical exemplars of a
15 Schayer, St., “Altindische Antizipationen der Aussagenlogik”, Bulletin international de
I’Academie Polonaise des Sciences et des Lettres, classe de philologies: 90-96 (1933); trans¬
lated in Jonardon Ganeri ed., Indian Logic: A Reader.
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category, and reasons that some new object belongs to the same category on the
grounds that it resembles in some appropriate and context determined manner one
of the exemplars. Models of case-based reasoning have been put forward for med¬
ical diagnostics and legal reasoning, and some have been implemented in artificial
intelligence models. It has been shown, for example, that training a robot to solve
problems by having it retrieve solutions to stored past cases, modifying them to
fit the new circumstances, can have great efficiency gains over seeking solutions
through the application of first principles. Perhaps something like this underlies a
lot of the way we actually reason, and perhaps it was as an attempt to capture this
type of reasoning that we should see the ancient logic of the Nyayasutra and indeed
of the medical theorist Caraka. 16 In this model, a perceived association between
symptoms in one case provides a reason for supposing there to be an analogous
association in other, resembling cases. The physician observing a patient A who
has, for example, eaten a certain kind of poisonous mushroom, sees a number of
associated symptoms displayed, among them F and G, say. He or she now en¬
counters a second patient B displaying a symptom at least superficially resembling
F. The physician thinks back over her past case histories in search of cases with
similar symptoms. She now seeks to establish if any of those past cases resembles
B, and on inquiry into B's medical history, discovers that B too has consumed
the same kind of poisonous mushroom. These are her grounds for inferring that B
too will develop the symptom G, a symptom that had been found to be associated
with F in A. A common etiology in the two cases leads to a common underlying
disorder, one that manifests itsself in and explains associations between members
of a symptom-cluster.
Can we find such a model of the informal logic of case-based reasoning in the
Nyayasutra ? Consider again NS 1.1.34. It said that ‘the reason is that which
proves what is to be proved in virtue of a similarity with the example.’ On our
reading, what this says is that a similarity between the symptom F in the new
case and a resembling symptom F' in the past-case or example is what grounds
the inference. And NS 1.1.36 says that ‘the example is something which, being
similar to that which is to be proved, has its character’. Our reading is that the
old case and the new share something in their circumstances, like having eaten
the same kind of poisonous mushroom, in virtue of which they share a ‘character’,
an underlying disorder that expains the clustering of symptoms. So the five-step
demonstration is now:
16 Caraka-Samhita 3.8.34: ‘what is called “example” is that in which both the ignorant and
the wise think the same and that which explicates what is to be explicated. As for instance, “fire
is hot,” “water is wet,” “earth is hard,” “the sun illuminates.” Just as the sun illuminates, so
knowledge of samkhya philosophy illuminates’.
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[thesis] Ga
[reason] Fa F is similar to F' in b.
[example] b exhibits the same underlying structure as a, because
it resembles a.
[application] a is the same as b with respect to G.
[conclusion] Ga
Let us see if this pattern fits examples of good inference taken from a variety
of early Indian logical texts. One pattern of inference, cited in the Nyayasutra,
is causal-predictive: Seeing the ants carrying their eggs, one infers that it will
rain; seeing a full and swiftly flowing river, one infers that it has been raining;
seeing a cloud of smoke, one infers the existence of an unseen fire. Presumably
the idea is that one has seen other ants carrying their eggs on a past occasion,
and on that occasion it rained. The inference works if, or to the exent that, we
have reasons for thinking that those ants and these share some unkown capacity,
a capacity that links detection of the imment arrival of rain with the activity of
moving their eggs. The pattern is similar in another kind of inference, inference
from sampling: Inferring from the salty taste of one drop of sea water that the
whole sea is salty; inferring that all the rice is cooked on tasting one grain. The
assumption again is that both the sampled grain of rice and any new grain share
some common underlying structure, a structure in virtue of which they exhibit the
sydromes associated with being cooked, and a structure whose presence in both
is indicated by their being in the same pan, heated for the same amount of time,
and so forth.
I will make two final comments about these patterns of case-based reason¬
ing. First, it is clear that background knowledge is essentially involved. As the
Nyayasutra stresses in its definition of a good debate, both parties in a debate
much be able to draw upon a commonly accepted body of information. Such
knowledge gets implicated in judgements about which similarities are indicative of
common underlying disorders, and which are not. Second, in such reasoning the
example does not seem to be redundant or eliminable in favour of a general rule.
For although there always will be a general law relating the underlying disorder
with its cluster of symptoms, the whole point of this pattern of reasoning is that
the reasoner need not be in a position to know what the underlying disorder is,
and so what form the general law takes. In conclusion, while the history of logic
in India shows a strong tendency towards formalisation, the logic of ancient In¬
dia tried to model informal patterns of reasoning from cases that are increasingly
becoming recognised as widespread and representative of the way much actual
reasoning takes place.
1.4 Refutation-only dialogue: vitanda
We have already seen how ‘refutation-only’ debate is defined in the Nyayasutra:
Refutation-only debate ( vitanda ) is one in which no counter-thesis is
proven. (NS 1.2.1-3).
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For the Naiyayika, to argue thus is to argue in a purely negative and destructive
way. Here one has no goal other than to undermine one’s opponent. People who
use reason in this way are very like the sceptics and unbelievers of the epics.
Vatsyayana claims indeed that to use reason in this way is virtually self-defeating:
A vaitandika is one who employs destructive criticism. If when ques¬
tioned about the purpose [of so doing], he says ‘this is my thesis’ or
‘this is my conclusion,’ he surrenders his status as a vaitandika. If he
says that he has a purpose, to make known the defects of the opponent,
this too will is the same. For if he says that there is one who makes
things known or one who knows, or that there is a thing by which
things are made known or a thing made known, then he surrenders his
status as a vaitandika. 17
Vitanda is the sceptic’s use of argumentation, and it is a familiar move to at¬
tempt to argue that scepticism is self-defeating. In India, it is the Madhyamika
Buddhist Nagarjuna (circa first century CE) who is most closely associated with
the theoretical elaboration of refutation-only argumentation, through the method
of ‘four-limbed refutation’ ( catuskoti) and the allied technique of presumptive rea¬
soning ( prasariga; tarka). In the next section, I will show how this latter technique
became a device for shifting the burden of proof onto one’s opponent. First, I
will examine the method of ‘four-limbed’ refutation in the context of Nagarjuna’s
wider philosophical project.
Reasoning, for Nagarjuna, is the means by which one ‘steps back’ from common
sense ways of understanding to a more objective view of the world. Reason is a
mode of critical evaluation of one’s conceptual scheme. A more objective under¬
standing is one in which one understands that things are not necessarily as they
appear. It is a view from which one can see how and where one’s earlier concep¬
tions are misleading. One learns not to trust one’s perceptions when a large object
far away looks small, or a stick half submerged in water looks bent, and in learning
this one exercises a mode of self-critical reason. So too a rational person learns
not to trust their conceptions when they presuppose the existence of independent,
self-standing objects. From the vantage point of an objective view, it is easy to see
that one’s old conceptions had false presuppositions. The real trick, however, is
to be able to expose those presuppositions while still ‘within’ the old conception,
and so to lever oneself up to a more objective view. This levering-up-from-within
requires a new way of reasoning: Nagarjuna’s celebrated prasanga-type rationality.
It is a self-critical rationality which exposes as false the existential presuppositions
on which one’s present conceptions are based. The same method can equally well
be used to expose the false presuppositions on which one’s dialectical opponents’
views are based, and for this reason the whole technique is strongly maieutic, in
the sense defined earlier.
A simple example will illustrate the kind of reasoning Nagarjuna thinks is needed
if one is to expose the presuppositions of one’s conceptual scheme from within. A
17
Nyayabhasya 3, 15-20.
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331
non-compound monadic concept ‘F’ has the following application-condition: it
applies only to things which are F. It is therefore a concept whose application
presupposes that there is a condition which divides the domain into two. For our
purposes, the condition can be thought of either as ‘belonging to the class of F s’
or ‘possessing the property being-F’. Now take an arbitrary object, a, from some
antecedently specified domain. There are apparently two possibilities for a: either
it falls under the concept, or else it is not. That is, the two options are:
(I) F applies to a
(II) F does not apply to a.
Suppose that one can disprove both of these options. How one would try to do
this will vary from case to case depending on the individual concept under scrutiny.
But if one is able to disprove (I) and to disprove (II), then the concept in question
can have no application-condition. The presupposition for the application of the
concept, that there is a condition (class, property) effecting a division within the
domain, fails. A later Madhyamika master 18 expresses the idea exactly:
When neither existence nor nonexistence presents itself before the
mind, then, being without objective support ( niralambana ) because
there is no other way, [the mind] is still.
Sentences are used to make statements, but if the statement so made is neither
true nor false, then, because there is no third truth-value, the statement must be
judged to lack content. 19
Nagarjuna’s developed strategy involves a generalization. A generalization is
needed because many if not most of the concepts under scrutiny are relational
rather than monadic, centrally: causes, sees, moves, desires. When a concept is
relational, there are four rather than two ways for its application-condition to be
satisfied (see Figure 1, page 332):
(I) R relates a only to itself
(II) R relates a only to things other than itself
(III) R relates a both to itself and to things other than itself
(IV) R relates a to nothing.
As an illustration of the four options, take R to the square-root relation yj, and
the domain of objects to be the set of real numbers. Then the four possibilities are
exemplified by the numbers 0, 4, 1 and -1 respectively. For y/0 = 0, y/4 = 2 and
also -2, y/1 = 1 and also -1, while finally -1 does not have a defined square root
among the real numbers. The list of four options is what is called in Madhyamaka
a catuskoti.
18 Santideva, Bodhicaryavatara 9.34.
19 On presupposition and truth-value gaps, see P. F. Strawson, Introduction to Logical Theory
(London: Methuen, 1952).
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Everything is thus, not thus, both thus and not thus, or neither thus nor
not thus. That is the Buddha’s [provisional] instruction. [ Mulamadhyanaka-
ka=arika , MK 18.8]
Some say that suffering ( duhkha ) is self-produced, or produced from
another, or produced from both, or produced without a cause. [MK
12 . 1 ]
Since every factor in existence ( dharma ) are empty, what is finite and
what is infinite? What is both finite and infinite? What is neither
finite nor infinite? [MK 25.22]
It is easy to see that the four options are mutually exclusive and jointly exhaus¬
tive. For the class of objects to which R relates a is either (IV) the empty set 0
or, if not, then either (I) it is identical to {a}, or (II) it excludes {a}, or (III) it
includes {a}. Not every relation exhibits all four options. (I) not exhibited if R
is anti-reflexive. (II) is not exhibited if R is reflexive and bijective. (IV) is not
exhibited if R is defined on every point in the domain. Note in particular that
if R is the identity relation, then neither (III) nor (IV) are exhibited, not (III)
because identity is transitive, and not (IV) because identity is reflexive. Indeed,
options (III) and (IV) are not exhibited whenever R is an equivalence (transitive,
symmetric, and reflexive) relation.
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333
The next step in the ‘refutation-only’ strategy is to construct subsidiary ‘dis¬
proofs’, one for each of the four options. Although there is no pre-determined pro¬
cedure for constructing such disproofs, by far the most commonly used method is to
show that the option in question has some unacceptable consequence ( prasanga ). I
will examine this method in detail in §1.5. A major dispute for later Madhyamikas
was over what sort of reasoning is permissible in the four subsidiary disproofs, the
proofs that lead to the rejection of each of the four options. It is a difficult question
to answer, so difficult indeed that it led, at around 500 AD, to a fission within
the school of Madhyamaka. The principal group (Prasangika, headed by Bud-
dhapalita) insisted that only prasanga- type, ‘presupposition-negating’ reasoning is
admissible. This faction is the more conservative and mainstream, in the sense that
their teaching seems to be in keeping with Nagarjuna’s own method of reasoning.
The important later Madhyamika masters Candraklrti and Santideva defended this
view. A splinter faction, however, (Svatantrika, headed by Bhavaviveka) allowed
‘independent’ inference or inductive demonstration into the disproofs. Perhaps this
was done so that the inferential methods developed by Dinnaga (§2.2) could be
deployed in establishing the Madhyamika’s doctrinal position. Clearly, the fewer
restrictions one places on the type of reasoning one permits oneself to use, the
greater are the prospects of successfully finding arguments to negate each of the
four options. On the other hand, we have seen that the citation of paradigmatic
examples is essential to this type of reasoning (§1.3), and it is hard to see how one
could be entitled to cite examples in support of one’s argument, when the very
conception of those examples is in question.
The effect of the four subsidiary disproofs is to establish that none of the four
options obtains: 20
Neither from itself nor from another, nor from both, nor without a
cause, does anything whatever anywhere arise. [MK 1.1]
One may not say that there is emptiness, nor that there is non-emptiness.
Nor that both, nor that neither exists; the purpose for so saying is only
one of provisional understanding. [MK 22.11]
The emptiness of the concept in question is now deduced as the final step in
the process. For it is a presupposition of one of the four options obtaining that
the concept does have an application-condition (a class of classes or relational
property), ff all four are disproved, then the presupposition itself cannot be true.
When successful, the procedure proves that the concept in question is empty, null,
sunya. This is Nagarjuna’s celebrated and controversial ‘ prasanga-type ’ rational
inquiry, a sophisticated use of rationality to annul a conceptual scheme.
20 Further examples: MK 25.17, 25.18, 27.15-18. Interesting is the suggestion of Richard
Robinson that the method of reasoning from the four options has two distinct functions, a
positive therapeutic role as exhibited by the unnegated forms, and a destructive dialectical role,
exhibited by the negated forms. Richard H, Robinson, Early Madhyamika in India and China ,
(Madison, Milwauke and London: University of Winsconsin Press, 1967), pp. 39-58, esp. pp.
55-56.
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A statement is truth-apt if it is capable of being evaluated as either true or
false. When Nagarjuna rejects each of the four options, he is rejecting the claim
that a statement of the form 1 aRb ’ is truth-apt, since the four options exhaust
the possible ways in which it might be evaluated as true. But if the statements
belonging to a certain discourse are not truth-apt, then the discourse cannot be
part of an objective description of the world (a joke is either funny or unfunny, but
it cannot be evaluated as true or false.) The prasanga negates a presupposition
for truth-aptness and so for objective reference.
Nagarjuna applies the procedure in an attempt to annul each of the concepts
that are the basic ingredients of our common-sense scheme. In each case, his
method is to identify a relation and prove that none of the four options can ob¬
tain. On closer inspection, it turns out that his argumentation falls into two basic
patterns. 21 One pattern is applied to any concept involving the idea of an ordering
or sequence, especially the concept of a causal relation, of a temporal relation and
of a proof relation. The paradigm for this argument is Nagarjuna’s presentation of
a paradox of origin (chapter 1), which serves as model for his analysis of causation
(chapter 8), the finitude of the past and future (chapter 11), and suffering (chapter
12). The argument seeks to establish that a cause can be neither identical to, nor
different from, the effect. If nothing within the domain is uncaused, then the four
options for the realization of a causal relation are foreclosed.
The other pattern of argumentation in Nagarjuna is essentially grammatical.
When a relational concept is expressed by a transitive verb, the sentence has an
Agent and a Patient (the relata of the relation). For example, “He sees the tree,”
“He goes to the market,” “He builds a house.” The idea of the grammatical argu¬
ment is that one can exploit features of the deep case structure of such sentences in
order to prove that the Patient can be neither identical to the Agent, nor include
it, nor exclude it, and that there must be a Patient. Nagarjuna uses this pattern
of argumentation in constructing a paradox of motion (MK, chapter 2), and this
chapter serves as a model for his analysis of perception (chapter 3), composition
(chapter 7), fire (chapter 10), and of bondage and release (chapter 16). Indeed,
the same pattern of argument seems to be applicable whenever one has a concept
which involves a notion of a single process extended in time. What exactly these
arguments show and how well they succeed is a matter of debate, but what we
have seen is the elaboration of a sophisticated sceptical strategy of argumentation,
based on the idea of ’refutation-only’ dialogue.
1.5 Presumptive argumentation ftarkaj and burden-of^proof shifting
Indian logicians developed a theory of what they call ‘suppositional’ or ’presump¬
tive’ argumentation ( tarka ). It is a theory about the burden of proof and the role
of presumption, about the conditions under which even inconclusive evidence is
sufficient for warranted belief. As we have already seen, it is a style of reason¬
ing that is regarded as permissible within a well-conducted dialogue ( vada ; see
21 On other patterns in Nagarjuna’s argumentation, see Robinson (1967: 48).
Indian Logic
335
§1.3). In the canonical and early literature, tarka is virtually synonymous with
reasoned thinking in general. The free-thinkers so derided in the epics were called
tarkikas or ‘followers of reason’. Even later on, when the fashion was to adorn
introductory surveys of philosophy with such glorious names as The Language of
Reason ( Tarkabhasa, , Moksakaragupta), Immortal Reason ( Tarkamrta. JagadTsa),
Reason’s Moonlight (Tarkakaumudi , Laugaksi Bhaskara), it was usual to confer on
a graduate of the medieval curriculum an honorific title like Master or Ford of
Reason ( tarkavagisa, tarkatirtha). Such a person is a master in the art of evidence
and the management of doubt, knowing when to accept the burden of proof and
also when and how to deflect it.
Extrapolative inference ( anumana , see §1.3) rests on the knowledge of universal
generalisations, and it is the possibility of such knowledge that the most troubling
forms of scepticism call into question. How can one be entitled to believe that
something is true of every member of a domain without inspecting each member
individually? How does one cope with the ineliminable possibility that an unper¬
ceived counterexample exists in some distant corner of the domain? The difficulty
here is with the epistemology of negative existentials. The Buddhist Dirinaga for¬
mulates the extrapolation relation as a ‘no counterexample’ relation. For him, x
extrapolates y just in case there is no x without y ( y-avina x-abhava ). The Navya-
Nyaya logicians prefer a different negative existential condition, one derived from
the reflexivity and transitivity of the extrapolation relation. Given transitivity, if
x extrapolates y then, for any z, if y extrapolates z, so does x. The converse of
this conditional holds too, given that the extrapolation relation is reflexive (proof:
let z — y). So let us define an ‘associate condition’ ( upadhi ) as a property which
is extrapolated by y but not x. Then x extrapolates y just in case there is no
associate condition. 22 One can infer fire from smoke but not smoke from fire, for
there is an associate condition, dampness-of-fuel, present wherever smoke is but
not wherever fire is. Tinkering with the definition, though, does not affect the
epistemological problem; it remains the one of proving a nonexistence claim.
Presumptive argumentation, tarka, is a device for appropriating a presumptive
right — the right to presume that one’s own position is correct even without con¬
clusive evidence in its support. One is, let us imagine, in a state of doubt as to
which of two hypotheses A and B is true. A and B are exclusive (at most one
is true) but not necessarily contradictory (both might be false). Technically, they
are in a state of ‘opposition’ ( virodha ). 23 The doubt would be expressed by an
exclusive disjunction in the interrogative - Is it that A or that B1 Uncertainty
initiates inquiry, and at the beginning of any inquiry the burden of proof is sym¬
metrically distributed among the alternative hypotheses. A piece of presumptive
22 For a survey of the literature on this theory, see Karl Potter ed., “Indian Epistemology and
Metaphysics”, Encyclopedia of Indian Philosophies , Volume 2 (Princeton: Princeton University
Press, 1977), pp. 203-206; Karl Potter and Sibajiban Bhattacharyya eds., “Indian Analytical
Philosophy: Gangesa to Raghunatha”, Encyclopedia of Indian Philosophies, Volume 6 (Prince¬
ton: Princeton University Press, 1993), pp. 187-192.
23 Nandita Bandyopadhyay, “The Concept of Contradiction in Indian Logic and Epistemology,”
Journal of Indian Philosophy 16.3 (1988), pp. 225-246, fn. 1.
336
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argumentation shifts the burden of proof by adducing a prima facie counterfactual
argument against one side. The form of the argument is the same in all cases. It is
that one alternative, supposed as true, would have a consequence in conflict with
some set of broadly defined constraints on rational acceptability. The existence of
such an argument gives one the right to presume that the other alternative is true,
even though one has no conclusive proof of its truth, and even though the logical
possibility of its being false remains open. In the psychologized language of the
Nyaya logician, a suppositional argument is a ‘blocker’ ( badhaka ) to belief in the
supposed alternative, and an ‘eliminator’ ( nirvartaka ) of doubt. The Naiyayika
Vacaspati (9th century) comments: 24
Even if, following a doubt, there is a desire to know [the truth], the
doubt still remains after the desire to know [has come about]. This
is the situation intended for the application of presumptive argumen¬
tation. Of two theses, one should be admitted as known when the
other is rejected by the reasoning called ‘suppositional.’ Thus doubt
is suppressed by the application of presumptive argumentation to its
subject matter... A means of knowing is engaged to decide a question,
but when there is a doubt involving its opposite, the means of knowing
fails [in fact] to engage. But the doubt concerning the opposite is not
removed as such by the undesired consequence. What makes possible
its removal is the means of knowing.
Vacaspati stresses that a thesis is not itself proved by a suppositional demon¬
stration that the opposite has undesired consequences; one still needs evidence
corroborating the thesis. But there is now a presumption in its favour, and the
burden of proof lies squarely with the opponent. Presumptive argumentation ‘sup¬
ports’ one’s means of acquiring evidence but it not itself a source of evidence. It
role is to change the standard of evidence required for proof in the specific context.
A radical sceptical hypothesis is a proposition inconsistent with ordinary belief
but consistent with all available evidence for it. The aim of the radical sceptic is
to undermine our confidence that our beliefs are justified, to introduce doubt. The
Nyaya logicians’ response to scepticism is not to deny that there is a gap between
evidence and belief, or to deny the logical possibility of the sceptical hypothesis.
It is to draw a distinction between two kinds of doubt, the reasonable and the
reasonless. A doubt is reasonable only when both alternatives are consistent with
all the evidence and the burden of proof is symmetrically distributed between them.
One paradigmatic example is the case of seeing in the distance something that
might be a person or might be a tree-stump. Udayana gives the epistemology
of such a case: it is a case in which one has knowledge of common aspects but
not of specific distinguishing features. What we can now see is that the example
gets its force only on the assumption that there is a level epistemic playing field,
with both hypotheses carrying the same prima facie plausibility. Presumptive
24 Nyayavarttikatatparyatika, below NS 1.1.40.
Indian Logic
337
argumentation has the potential to break the impasse — imagine, for example,
that the unidentified lump is just one of ten in an orderly row not there an hour
ago. The perceptual evidence remains the same, but the burden of proof is on
anyone who wants to maintain in this situation that the lump is a stump.
The other paradigm is knowledge of extrapolation relations. The problem here
is that the thesis is one of such high generality that the burden of proof is al¬
ready heavily against it! How can a few observations of smoke with fire ground
a belief that there is fire whenever there is smoke? Suppositional argument has
a different supportive role here. Its function is to square the scales, to neutralise
the presumption against the belief in generality. It does so by finding prima facie
undesirable consequences in the supposition that an associate condition or coun¬
terexample exists. Then sampling (observation only of confirmatory instances in
the course of a suitably extensive search for counterexamples), though still weak
evidence, can tilt the scale in its favour.
A presumptive argument moves from conjecture to unacceptable consequence.
Modern writers often identify it with the medieval technique of reductio ad ab-
surdum, but in fact its scope is wider. The ‘unacceptable consequence’ can be an
out-and-out contradiction but need not be so. For we are not trying to prove that
the supposition is false, but only to shift the burden of proof onto anyone who
would maintain it. And for this it is enough simply to demonstrate that the sup¬
position comes into conflict with some well-attested norm on rationality. Udayana,
the first to offer any systematic discussion, does not even mention contradiction as
a species of unacceptable consequence. He says 25 that presumptive argumentation
is of five types -
1. self-dependence ( atmasraya )
2. mutual dependence ( itaretarasraya )
3. cyclical dependence ( cakraka )
4. lack of foundation ( anavastha )
5. undesirable consequence ( anistaprasahga)
The last of these is really just the generic case, what distinguishes presump¬
tive argumentation in general. The first four form a tight logical group. If the
supposition is the proposition A, then the four types of unacceptable consequence
are (1) proving A from A, (2) proving A from B, and B from A, (3) proving A
from B, B from C, and C from A — or any higher number of intermediate proof
steps eventually leading back to A, and (4) proving A from B from C, C from
D,.. without end. So what presumptive argumentation must show is that the
supposition is ungrounded, its proof being either regressive or question-begging.
Two points are noteworthy about Udayana’s list. First, rational unacceptability
bears upon the proof adduced for the supposition, not the supposition itself. The
25 Atmatattvaviveka, p. 863.
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underlying implication is that one has the right to presume that one’s thesis is
correct if one can find fault with the opponent’s proof of the antithesis. Principles
of this sort are familiar from discussion of the informal logic of arguments from
ignorance in which one claims entitlement to assert A on the grounds that it is
not known (or proved) that ->A. 26 In general such a claim must be unfounded - it
amounts to the universal appropriation of a presumptive right in all circumstances.
The second point to notice about Udayana’s list, however, is that it is very
narrow. Udayana places strict constraints on what will count as an unacceptable
consequence, constraints which are more formal than broadly rational. Conflict
with other well-attested belief is not mentioned, for instance. Udayana severely
limits the scope of presumptive argumentation. His motive, perhaps, is to disarm
the sceptic. For presumptive argumentation is the favoured kind of reasoning
of the sceptic-dialecticians (and indeed the term Udayana uses is prasariga, the
same term Nagarjuna had used for his dialectical method). Sceptics typically will
want to loosen the conditions on what constitutes an unacceptable consequence
of a supposition, so that the scope for refutation is expanded. So what Udayana
seems to be saying is that one does indeed have the right to presume that one’s
thesis is correct when the argument for the counter-thesis commits a fallacy of a
particularly gross type — not mere conflict with other beliefs but formal lack of
foundation. If the best argument for the antithesis is that bad, then one has a
prima facie entitlement to one’s thesis.
Srlharsa (c. AD 1140) is an Advaita dialectician, a poet and a sceptic. 27 He
expands the notion of unacceptable consequence, noticing several additional types
unmentioned by Udayana. 28 One is ‘self-contradiction’ ( vyaghata ). It was Udayana
himself 29 who analysed the notion of opposition as noncompossibility, and cited
as examples the statements “My mother is childless,” “I am unable to speak”, and
“I do not know this jar to be a jar.” In the first instance, the noncompossibility
is in what the assertion states, in the second it is in the speech-act itself, while in
the third the propositional attitude self-ascription is self-refuting (a case akin to
the Cartesian impossibility of thinking that one is not thinking).
Another refutation-exacting circumstance is the one called ‘recrimination’ ( prat-
ibandi). This is a situation in which one’s opponent accuses one of advancing a
faulty proof, when his own proof suffers exactly the same fault! There is a disagree¬
ment about what this state of equifallaciousness does to the burden of proof. The
26 Douglas Walton, Arguments from Ignorance (University Park, PA: The Pennsylvania State
University Press, 1996).
27 On Srlharsa: Phyllis Granoff, Philosophy and Argument in Late Vedanta: Sriharsa’s
Khandanakhandakhadya (Dordrecht: Reidel Publishing Company, 1978); Stephen Phillips, Clas¬
sical Indian Metaphysics (La Salle: Open Court, 1995), chapter 3.
28 Khandanakhandakhadya IV, 19 ( aprasangatmakatarkanirupana , pp. 777-788, 1979 edition;
section numbering follows this edition). SrTharrsa the negative dialectician wants to criticise even
the varieties of presumptive argumentation, although his own method depends upon it. So he
says: “By us indeed were presumptive argumentations installed in place, and so we do not reject
them with [such] counter-arguments. As it is said - ‘it is wrong to cut down even a poisonous
tree, having cultivated it oneself’” (p. 787).
29 Atmatattvaviveka, p. 533.
Indian Logic
339
practice of Naiyayikas is to take the circumstance as tilting the balance against
the opponent - the opponent discredits himself in pressing an accusation without
seeing that it can be applied with equal force to his own argument. But SrTharsa
quotes with approval Kumarila’s assertion that “all things being equal, where the
same fault afflicts both positions one should not be censured [and not the other]” . 30
SrTharsa, the sceptic, would like to see both parties refuted by this circum¬
stance. The same point underlies his mention as an unacceptable consequence the
circumstance of ‘lack of differential evidence’ (vinigamanaviraha) , when thesis and
antithesis are in the same evidential situation. Again, what we see is a jostling
with the burden of proof. Here SrTharsa is saying that absence of differential evi¬
dence puts a burden of proof on both thesis and antithesis — doubt itself refutes.
It is the sceptic’s strategy always to seek to maximise the burden of proof, and
so to deny that anyone ever has the right to presume their position to be correct.
That is, as Stanislaw Schayer observed a long time ago, a difference between the
tarka of the Naiyayika and the prasanga of a sceptic like SrTharsa or Nagarjuna. 31
For the latter, the demonstration that a thesis has an allegedly false consequence
does not commit the refuter to an endorsement of the antithesis. Nagarjuna wants
to maintain instead that thesis and antithesis share a false existential precommit¬
ment.
Simplicity ( laghutva ) is, SrTharsa considers and the Naiyayikas agree, a ceteris
paribus preference-condition. Of two evidentially equivalent and otherwise ratio¬
nally acceptable theses, the simpler one is to be preferred. The burden of proof
lies with someone who wishes to defend a more complex hypothesis when a simpler
one is at hand. The Nyaya cosmological argument appeals to simplicity when it
infers from the world as product to a single producer rather than a multiplicity of
producers. Here too the role of the simplicity consideration is to affect the burden
of proof, not itself to prove. Cohen and Nagel 32 make a related point when they
diagnose as the ‘fallacy of simplism’ the mistake of thinking that “of any two hy¬
potheses, the simpler is the true one.” In any case, simplicity can be a product
not of the content of a hypothesis but only of its mode of presentation — the
distinction is made by the Naiyayikas themselves. 33 And it is hard to see how it
can be rational to prefer one hypothesis to another only because it is simpler in
form.
We have assumed that the rival hypotheses are both empirically adequate, that
is to say, they are both consistent with all known facts. SrTharsa mentions an
unacceptable consequence involving empirical evidence ( utsarga ). It is an objec-
30 Khandanakhandakhadya II, 2 ( pratibandilaksanakhandana , pp. 571-572). The full quotation
is given in his commentary by Samkara Misra.
31 Stanislaw Schayer, “Studies on Indian Logic, Part II: Ancient Indian Anticipations of Propo¬
sitional Logic,” [1933], translated into English by Joerg Tuske in Jonardon Ganeri ed., Indian
Logic: A Reader.
32 Morris R. Cohen and Ernest Nagel, An Introduction to Logic and Scientific Method (London:
Routledge & Kegan Paul, 1934), p. 384.
33 Bhimacarya Jhalakikar, Nyayakosa or Dictionary of Technical Terms of Indian Philosophy
(Poona: Bhandarkar Oriental Research Institute, 1928), s.v. laghutvam.
340
Jonardon Ganeri
tion to the usual idea that if there is empirical evidence supporting one hypothesis
but not the other, then the first is confirmed. Srlharsa’s sceptical claim is that a
hypothesis must be considered refuted unless it is conclusively proved; nonconclu-
sive empirical evidence does nothing to affect this burden of proof. Likewise, he
says, a hypothesis must be considered refuted if it is incapable of being proved or
disproved — this at least seems to be the import of the unacceptable consequence
he calls ‘impertinence’ ( anucitya ) or ‘impudence’ ( vaiyatya).
Other varieties of suppositional refutation have been suggested along lines simi¬
lar to the ones we have reviewed. Different authors propose different sets of criteria
for rational nonacceptance. What we have seen is that there is, in the background,
a jostling over the weight and place of the burden of proof. The sceptic presses
in the direction of one extreme — that a thesis can be considered refuted unless
definitively proven. The constructive epistemologist tries to press in the direction
of the opposite extreme — that a thesis can be considered proved unless definitively
disproved. The truth lies somewhere in between, and it is the role of presumptive
argumentation to locate it.
2 BUDDHIST CONTRIBUTIONS IN INDIAN LOGIC: FORMAL CRITERIA
FOR GOOD ARGUMENTATION
2.1 The doctrine of the triple condition (trairupya)
The Buddhist logician Dinnaga (c. 480-540 AD) recommends a fundamental re¬
structuring of the early Nyaya analysis of reasoned extrapolation and inference.
Recall that analysis. It is an inference from likeness and unalikeness. In the one
case, some object is inferred to have the target property on the grounds that it is
‘like’ a paradigmatic example. The untasted grain of rice is inferred to be cooked
on the grounds that it is in the same pan as a test grain which is found to be
cooked. In the other case, the object is inferred to have the target property on the
grounds that it is ‘unlike’ an example lacking the target property. Likeness and un¬
alikeness are matters of sharing or not sharing some property, the reason-property
or evidence grounding the inference. Examples are either ‘positive’ — having both
the reason and the target property, or ‘negative’ — lacking both. Extrapolation
is the process of extrapolating a property from one object to another on the basis
of a likeness or unalikeness between them.
The difficulty is that not every such extrapolation is rational or warranted. The
extrapolation of a property from one object to another is warranted only when
the two objects are relevantly alike or relevantly unalike. That two objects are
both blue does not warrant an extrapolation of solidity from one to the other;
neither can we infer that they are different in respect to solidity because they
are of different colours. What one needs, then, is a theory of relevant likeness or
unalikeness, a theory, in other words, of the type of property (the reason property)
two objects must share if one is to be licensed to extrapolate another property (the
target property) from one to the other.
Indian Logic
341
This is exactly what Diiinaga gives in his celebrated theory of the ‘reason with
three characteristics’ ( trairupya ). Dihnaga’s thesis is that relevant likeness is an
exclusion relation. Two objects are relevantly alike with respect to the extrap¬
olation of a property 5 just in case they share a property excluded from what
is other than S. In other words, a reason property H for the extrapolation of
a target property S is a property no wider in extension than 5 (assuming that
‘non’ is such that HCi nonS = 0 iff H C S). Here is the crucial passage in the
Pramana-samuccaya, or Collection on Knowing:
The phrase [from II lb] “through a reason that has three characteris¬
tics” must be explained.
[A proper reason must be] present in the site of inference and
in what is like it and absent in what is not [II 5cd].
The object of inference is a property-bearer qualified by a property. Af¬
ter observing [the reason] there, either through perception or through
inference, one also establishes in a general manner [its] presence in
some or all of the same class. Why is that? Because the restriction
is such that [the reason] is present only in what is alike, there is no
restriction that it is only present. But in that case nothing is accom¬
plished by saying that [the reason] is “absent in what is not”. This
statement is made in order to determine that [the reason], absent in
what is not [like the site of inference], is not in what is other than or
incompatible with the object of inference. Here then is the reason with
three characteristics from which we discern the reason-bearer.
Dihnaga’s important innovation is to take the notions of likeness and unalike-
ness in extrapolation to be relative to the target property rather than the reason
property. Two objects are ‘alike’ if they both have, or both lack, the target prop¬
erty. Two objects are ‘unalike’ if one has and the other lacks the target property.
We want to know if our object — the ‘site’ of the inference — has the target
property or not. What we do know is that our object has some other property, the
reason property. So what is the formal feature of that reason property, in virtue
of which its presence in our object determines the presence or absence of the tar¬
get property? The formal feature, Dinnaga claims, is that the reason property is
present only in what is alike and absent in whatever is unalike our object.
This can happen in one of two ways. It happens if the reason property is absent
from everything not possessing the target property and present only in things
possessing the target property. Then we can infer that our object too possesses
the target property. It can also happen if the reason property is absent from
everything possessing the target property and present only in things not possessing
the target property. Then we can infer that our object does not possess the target
property.
Call the class of objects which are like the site of the inference the ‘likeness class’,
and the class of objects unlike the site the ‘unlikeness class’ (Dihnaga’s terms are
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sapaksa and vipaksa). Interpreters have traditionally taken the likeness class to be
the class of objects which possess the target property, and the unlikeness class to
be the class of objects which do not possess the target property. I read Dinnaga
differently. I take his use of the terms ‘likeness’ and ‘unlikeness’ here at face-value,
and identify the likeness class with the class of things in the same state vis-a-vis
the target property as the site of the inference. We do not know in advance what
that state is, but neither do we need to. The pattern of distribution of the reason
property tells us what we can infer - that the site has the target property, that
it lacks it, or that we can infer nothing. My approach has several virtues, chief
among which is that it preserves the central idea of likeness as a relation between
objects rather than, as with the traditional interpretation, referring to a property
of objects. I think it also avoids many of the exegetical problems that have arisen
in the contemporary literature with regard to Dinnaga’s theory.
One of the traditional problems is whether the site of the inference is included
in the likeness class or not. 34 If the likeness class is the class of objects possessing
the target property, then to include it seems to beg the question the inference is
trying to resolve: does the site have that property or not. But to exclude it implies
that the union of the likeness and unlikeness classes does not exhaust the universe
(the site cannot, for obvious reasons, be unlike itself). So one is left with two
disjoint domains, and an apparently insuperable problem of induction - how can
correlations between the reason property and the target property in one domain
be any guide to their correlation in another, entirely disjoint, domain? 35
If we take Dinnaga’s appeal to the idea of likeness at face-value, however, the
problem simply does not arise. The site of the inference is in the likeness class on
the assumption that likeness is a reflexive relation — but that begs no question, for
we do not yet know whether the likeness class is the class of things which possess
the target property, or the class of things which do not possess it. It is the class of
things which are in the same state vis-a-vis the target property as the inferential
site itself. We can, if needs be, refer to objects ‘like the site but not identical
to it;’ or we can take likeness to be nonreflexive, and refer instead, if needs be,
to ‘the site and objects like it’ — but this is a matter only of labelling, with no
philosophical interest.
Another of the traditional problems with Dinnaga’s account is an alleged logical
equivalence between the second and third conditions. 36 The second condition
states that the reason property be present only in what is alike. 37 The third
condition states that it be absent in what is not. But if it is present only in what
34 Tom F. Tillemans, “On sapaksa,” Journal of Indian Philosophy 18 (1990), pp. 53-80.
35 Hans H. Herzberger, “Three Systems of Buddhist Logic,” in B. K. Matilal and R. D. Evans
eds., Buddhist Logic and Epistemology: Studies in the Buddhist Analysis of Inference and Lan¬
guage (Dordrecht: Reidel Publishing Company, 1982), pp. 59-76.
36 Bimal Matilal, “Buddhist Logic and Epistemology,” in Matilal and Evans (1982: 1-30);
reprinted in Matilal, The Character of Logic in India (Albany: State University Of New York
Press, 1998), chapter 4.
37 There is some debate among scholars over whether it was Dinnaga himself or his commenator
DharmakTrti who first inserts the particle only into the clauses.
Indian Logic
343
is alike, it must be absent in what is not; and if it is absent in what is not alike, it
must be present only in what is. Now it is clear that Dihnaga’s reason for inserting
the particle only into his formula is to prevent a possible misunderstanding. The
misunderstanding would be that of taking the second condition to assert that the
reason property must be present in all like objects. That would be too strong
a condition, ruling out any warranted inferences in which the reason property is
strictly narrower than the target. On account of the meaning of the particle only ,
we can see that it is also one of the two readings of the statement:
In what is alike, there is only the presence [of the reason]
where the particle only is inserted into the predicate position. Dinnaga eliminates
this unwanted reading of the second condition, but he does so in a disastrous way.
He eliminates it by inserting the particle into the subject position:
Only in what is alike, there is the presence [of the reason].
The reason this is disastrous is that it makes the second condition logically
equivalent to the third. Notice, however, that when only is in predicate position,
there are still two readings. The reading one needs to isolate is the second of these
two readings:
In what is alike, there is indeed the presence [of the reason]
That is, the reason is present in some of what is alike.
Accordingly, the theory is this. The extrapolation of a property S to an object
is grounded by the presence in that object of any property X such that X excludes
nonS but not S. A reason property for S is any member of the class
{X :XnS ^0 k XO nonS — 0}.
The clause ‘but not S’ (the second of Dihnaga’s three conditions) has a clear
function now. It is there to rule out properties which exclude both nonS and S.
Such properties are properties ‘unique’ to the particular object which is the site of
the inference, and Dinnaga does not accept as warranted any extrapolation based
on them. I will look at his motives in the next section.
Reason properties are nonempty subsets of the properties whose extrapolation
they ground. If two objects are ‘alike’ in sharing a property, and one has a second
property of wider extension than the first, then so does the second. Inductive
extrapolation, in effect, is grounded in the contraposed universal generalisation
“where the reason, so the target.” A difficult problem of induction remains - how
can one come to know, or justifiably believe, that two properties stand in such a
relation without surveying all their instances? Dinnaga has no adequate answer
to this problem (but see [Tuske, 1998; Peckhaus, 2001]). DharmakTrti, Dihnaga’s
brilliant reinterpreter, does. His answer is that when the relation between the
two properties is one of causal or metaphysical necessity, the observation of a few
344
Jonardon Ganeri
instances is sufficient to warrant our belief that it obtains (§2.3). Diinaga, however,
is not interested in such questions. For him, the hard philosophical question is that
of discovering the conditions for rational extrapolation. It is another issue whether
those conditions can ever be known to obtain.
To sum up, Diiinaga’s three conditions on the reason are:
1. Attachment Presence in the site a attachment ( paksadharmata )
2. Association Presence (only) in what is like ( anvaya )
3. Dissociation Absence in what is unlike ( vyatireka)
If we take these conditions to be independent, it follows that there are exactly
seven kinds of extrapolative inferential fallacy — three ways for one of the con¬
ditions to fail, three ways for two conditions to fail, and one way for all three
conditions to fail. So the new theory puts the concept of a fallacy on a more
formal footing. A fallacy is no longer an interesting but essentially ad hoc maxim
on reasoned argument. It is now a formal failing of the putative reason to stand
in the correct extrapolation-grounding relation. One way for the reason to fail
is by not attaching to the site at all, thereby failing to ground any extrapola¬
tion of other properties to it. This is a failure of the first condition. Another
way for the reason to fail is by ‘straying’ onto unlike objects, thereby falsifying
the third condition. The presence of one property cannot prove the presence of
another if it is sometimes present where the other one is not. (It can, however,
prove the absence of the other if it is only present where the other is not — and
then the absence of the first property is a proof of the presence of the second.)
We might then think of the third condition as a ‘no counter-example’ condition,
a counter-example to the extrapolation-warranting relation of subsumption being
an object where the allegedly subsumed property is present along with the absence
of its alleged subsumer. An extrapolation is grounded just as long as there are no
counterexamples.
2.2 Dinnaga’s 'wheel of reasons’ (hetucakra)
In addition to his Pramana-samuccaya , Dihnaga wrote another, very brief text
on logic, the Wheel of Reasons, or Hetucakranirnaya. Dinnaga’s aim here is to
classify all the different types of argument which fit into the general schema (p has
s because it has h), and to give an example of each. It is here that he applies his
theory of a triple-conditioned sign to show when an inference is sound or unsound,
and the kinds of defect an inferential sign can suffer from. Hence, it leads to a
classification of fallacious and non-fallacious inferences.
The ‘wheel’ or ‘cycle’ is in fact a 3 by 3 square, giving nine inference types.
Dihnaga derives the square as follows. A ‘homologue’ ( sapaksa ) is defined as any
object (excluding the locus of the inference) which is possesses the inferrable prop¬
erty, s. Now, a putative inferential sign, h, might be either (i) present in every
homologue, (ii) present in only some of the homologues but not in others, or (iii)
present in no homologue. Suppose we let ‘sp’ stand for the class of homologues.
Indian Logic
345
Then we can represent these three possibilities as ‘sp+\ ‘sp±\ and ! sp-’ respec¬
tively. The same three possibilities are also available with respect to the class of
heterologues (objects, excluding the locus, which do not possess the inferred prop¬
erty, s). We can denote these by ‘vp+\ ‘vp±’, and ‘vp-’ respectively. Thus, ‘vp+’
means that every member of vp (every heterologue) possesses the sign property,
h, etc. Now since any putative inferential sign must either be present in all, some
or no homologue, and also in either all, some or no heterologue, there are just nine
possibilities (Figure 2):
vp
+ - ±
+
1
2
3
deviating
goodK
deviating
sp -
4
5
6
contradictory
uniquely
contradictory
deviating
±
7
8
9
deviating
good
deviating
Figure 2.
Why does Dinnaga say that only 2 and 8 are cases of a good inferential sign?
Recall the three conditions on a good sign. The first is that the inferential sign
must be present in the locus of inference. This is taken for granted in the wheel.
The second states that the inferential sign should be present in some (at least
one) homologous case. In other words, a good sign is one for which either ‘sp+’
or ‘sp±’. Thus the second condition rules out 4, 5 and 6. Similarly, the third
condition states that the inferential sign should be absent from any heterologous
case, i.e. that ‘vp—’. This rules out 1, 4, 7 and 3, 6, 9. So only 2 and 8 represent
inferential signs which meet all three conditions and generate good inferences. Note
here that the third condition alone is sufficient to rule out every fallacious case
except 5. Hence, seeing why Dinnaga considers ‘type-5’ inferences to be unsound
will reveal why he considered the second of the three conditions to be necessary
(see below).
Dirinaga next gives an illustration of each of the nine possibilities. They can be
tabulated, as in Figure 3.
In each case, the locus of the inference is sound. Note that wherever possible,
Dinnaga cites both a ‘positive confirming example’, i.e. an object where both
346
Jonardon Ganeri
s
h
positive
example
negative
example
counter¬
example
1
eternal
knowable
space
—
a pot
2
transitory
created
a pot
space
—
3
manmade
tansitory
a pot
space
lightning
4
eternal
crated
—
—
a pot
5
eternal
audible
—
a pot
—
6
eternal
manmade
—
lightning
a pot
7
natural
transitory
lightning
—
a pot
8
transitory
manmade
a pot
space
—
9
eternal
incorporeal
space
a pot
action
Figure 3.
h and s are present, as well as a negative confirming example’, i.e. an object
where neither h nor s is present. Both support the inference. He also cites, where
relevant, a ‘counter-example’, i.e. a case where h is present but s is absent. The
existence of a counter-example undermines the inference. Let us look at four
representative cases.
Case 2: A warranted inference. This inference reads: Sound is transitory, be¬
cause it is created, e.g. a pot; space. Intuitively, this inference is sound, because the
reason-property, createdness, is present only in places where the inferred property,
transitoriness, is also present. Hence createdness is a good sign of transitoriness.
The inference is supported first by an example where both are present, a pot, and
second by an example where neither are present, space.
Case 3: ‘deviating’ ( asiddha ). This inference reads: Sound is manmade, because
it is transitory, e.g. a pot; space. Intuitively, this inference is unsound, because
the reason-property, transitoriness, is present in places where the inferred property,
manmade, is absent. The counterexample cited is lightning — transitory but not
manmade. Because we can find such a counter-example, the inferential sign is said
to ‘deviate’ from the inferred property. Deviating inferences are ones which satisfy
the second condition but fail the third.
Case 6: ‘contradictory’ ( viruddha ). The inference reads: Sound is eternal, be¬
cause it is manmade, e.g. lightning. The sign here fails both conditions 2 and
3 — there is no case of a thing which is eternal and manmade, but there is a
counter-example, for instance, a pot, which is manmade but non-eternal. Such
an inference is called ‘contradictory’ because we can in fact infer to the contrary
conclusion, namely that sound is non-eternal because it is manmade. We can do
this because in the contrary inference, the homologous and heterologous domains
are switched round.
Case 5: ‘specific’ ( asadharana ). Sound is eternal, because it is audible, e.g.
a pot. The first point to notice is that there are no counter-examples to this
Indian Logic
347
inference, for there are no examples, outside the ‘locus’ domain of sounds, of an
audible thing which is non-eternal. This is because there are no audible things
other than sounds! Hence the third condition seems to be satisfied trivially. The
characteristic of type-5 inferences is that the reason-property is ‘unique’ to the
locus. According to Dinnaga, such inferences are unsound, and the reason is that
they fail the second condition - there is no homologue, i.e. an eternal thing other
than sound, which is also audible.
But this just restates the characteristic feature of such inferences, it doesn’t
explain why they are unsound. Some modern authors argue that the significance
of the second condition is more epistemological, than logical: the second condi¬
tion implies that there must be a positive supporting example, and without such
an example the inference, even if sound, carries no conviction. Dinnaga might,
however, have had a more formal or logical reason for rejecting type-5 inferences.
The universal rule here is “Whatever is audible, apart from sound, is eternal”.
Now if a universal rule of the form ‘(Vx)(Fx Gx)’ is made true by there being
no F s, then so is the rule ‘(Vx)(Fx —> not-Gx)’. Hence, we could equally infer
that sound is non-eternal because it is audible! This resembles the fault which the
Nyaya called ‘ prakaranasama' or ‘indecisive’. Dinnaga, it seems, wants to avoid
this by saying that ‘(Vx)(Fx —► Gx)' is true only if there is at least one F, which
leads to the second condition.
Let us consider the argument, from specifics further. I have said that an extrap¬
olation-grounding property is a nonempty subproperty — a property narrower in
extension than the property being extrapolated, and resident at least in the ob¬
ject to which that property is being extrapolated. The sweet smell of a lotus is a
ground for extrapolating that it has a fragrance; its being a blue lotus is a ground
for extrapolating its being a lotus. Extrapolation is a move from the specific to the
general, from species to genus, from conjunction to conjunct. Extrapolation is a
move upwards in the hierarchy of kinds. This model of extrapolation works well in
most cases, but what happens at the extremes? The extreme in one direction is a
most general property of all, a property possessed by everything. Existence or ‘re¬
ality’, if it is a property, is a property like this, and the theory entails that existence
is always extrapolatable — the inference ‘o is, because a is F’ is always warranted.
Dinnaga’s theory faces a minor technical difficulty here. Since everything exists,
then everything is ‘like’ the site of the inference (in the same state as the site
with respect to existence), and the unlikeness class is empty. So Dinnaga has to
be able to maintain that his third condition — absence of the reason property in
every unlike object — is satisfied when there are no unlike objects. The universal
quantifier must have no existential import. His innovative distinction between in¬
ference ‘for oneself’ ( svarthanumdna ) and inference ‘for others’ ( pararthanumana )
is a help here. It is the distinction between the logical preconditions for warranted
extrapolation and the debate-theoretic exigencies of persuasion. While it might
be useful, even necessary, to be able to cite a supporting negative example if one’s
argument is to carry conviction and meet the public norms on believable inference,
348
Jonardon Ganeri
there is no corresponding requirement that the unlikeness class be nonempty if an
extrapolation is to be warranted.
What happens at the other extreme? Extrapolation is a move from the more
specific to the less specific, and the limit is the case when the reason property is
entirely specific to the site of the inference. There is no doubt but that Dinnaga
thinks that extrapolation breaks down at this limit. He calls such reason proper¬
ties ‘specific indeterminate’ (asadharananaikantika) , and classifies them as bogus-
reasons. Indeed it is the entire function of his second condition to rule out such
properties. That is why the second condition insists that the reason property must
be present in an object like the site. This condition is an addition to the first,
that the reason property be present in the site — it demands that the reason be
present in some other object like but not identical to the site. Diiinaga’s example
in the Collection on Knowing [II 7d] is:
[Thesis] Sound is noneternal.
[Reason] Because it is audible.
In the Wheel of Reasons [5cd-7a], he gives another example:
[Thesis] Sound is eternal.
[Reason] Because it is audible.
What is the difference? In fact, the difference between these two examples holds
the key to what Dinnaga thinks is wrong. The property audibility, something
specific to sound, does not determine whether sound is eternal or noneternal. In
either case, audibility is absent from what is unlike sound (because it is unique to
sound) but also from what is like sound (except for sound itself). This symmetry
in the distribution of the reason property undermines its capacity to discriminate
between truth and falsity. To put it another way, if we take the universal quantifier
to range over everything except the site of the inference, sound, then it is true both
that everything audible is eternal and that everything audible is noneternal - both
are true only because there are no audibles in the range of the quantifier.
This seems to be Dirinaga’s point, but it is not very satisfactory. Sound is either
eternal or noneternal, and so audibility is a subproperty of one or the other. One
and only one of the above universal quantifications is true when the quantifier is
unrestricted. In any case, just why is it that we should not reason from the specific
properties of a thing? We do it all the time. Historical explanations are notoriously
singular — unrepeated historical events are explained by specific features of their
context. Dinnaga, it seems, is like the follower of the deductive-nomological model
in insisting on repeatability as a criterion of explanation. What about mundane
cases like this one: the radio has stopped because I have unplugged it? Being
unplugged by me is a property specific to the radio, and yet the form of the
explanation seems unapproachable. Perhaps, however, what one should say is that
the explanatory property is ‘being unplugged’, and not ‘being unplugged by me’,
Indian Logic
349
and the explanation rests on the generalisation ‘whenever a radio is unplugged, it
stops.’ So then the restriction is not to any property specific to the site, but only
to those which are not merely tokens of some more general explanatory property.
And yet there are still intuitively rational but specific inferences — that salt is
soluble because it has a certain molecular structure, that helium is inert because
it has a certain atomic number, flying creatures fly because they have wings. Why
shouldn’t the specific properties of a thing be implicated in inferences of its other
properties?
What we see here is Dihnaga’s adherence to a strictly inductivist model of
extrapolation. The specific property audibility does not ground an extrapolation
of eternality or noneternality because there can be no inductive evidence for the
extrapolation. Inductive evidence takes the form of objects in the likeness and
unlikeness classes known to have or not to have the reason. One might think that
one does have at least ‘negative’ evidence, for one knows that audibility is absent
from any object in the unlikeness class. So why can one not infer from the fact
that audibility is absent in unlike objects that it must be present in like objects?
The answer is that one can indeed make that inference, but it does not get one
very far. For we must recall again the way these classes are defined - as classes of
objects like or unlike the site with respect to eternality. We do not know whether
the site is eternal or noneternal, and in consequence we do not know whether
unlike things are things which are noneternal or eternal. So while we have plenty
of examples of eternal inaudibles and noneternal inaudibles, we still do not know
which are the ‘alike’ ones and which the ‘unalike’.
The explanation of salt’s solubility by its specific molecular structure exemplifies
a quite different model of explanation. It is a theoretical explanation resting on
the postulates of physical chemistry. It is from theory, not from observation, that
one infers that having an NaCl lattice structure is a subproperty of being soluble.
Similarly, within the context of suitable theories about the nature of sound and
secondary qualities, one might well be able to infer from sound’s being audible to
its being noneternal. Diiinaga, in spite of his brilliance and originality, could not
quite free himself from the old model of inference from sampling. His inclusion
of the second condition was a concession to this old tradition. He should have
dropped it. Later Buddhists, beginning with Dharmaklrti, did just that - they
effectively dropped the second condition by adopting the reading of it that makes
it logically equivalent to the third.
Dihnaga’s insistence that any acceptable inference should be accompanied by
both positive and negative supporting examples provoked the Naiyayika Uddy-
otakara to criticise and expand the Wheel. Uddyotakara points out that there are
sound patterns of inference in which either the class of homologues or the class of
heterologues is empty. These he calls the ‘universally negative’ ( kevala-vyatikekin )
and ‘universally positive’ ( kevalanvayin ) inferences. We now have a wheel with
sixteen possible cases (Figure 4):
Here, ‘o’ means that the class (sp or vp) is empty. An example of a sound
‘universally positive’ inference might be: “This exists because I can see it”. There
350
Jonardon Ganeri
vp
+ - ± o
+
sp -
±
0
good
good
good
good
good
Figure 4.
are no heterologues, because there are no things which do not exist, and so there
are no negatively supporting examples. Nevertheless, we should recognise the
acceptibility such an inference. Examples of ‘universally negative’ inferences are
more difficult to find. The later Nyaya link such inferences with their theory of
definition, considering such examples as “Cows are distinct from non-cows, because
they have dewlap’. There are no objects which are distinct from non-cows except
for cows, and hence no homologues. But the inference might have significance,
for it tells us that the property of having dewlap serves to distinguish cows from
non-cows, and hence can be used as a definition of cowhood 38 .
2.3 Arguments from effect, essence and non-observation
Dharmaklrti (AD 600-660) offers a substantive account of the conditions under
which the observation of a sample warrants extrapolation. His claim is that this
is so if the reason property is one of three types: an ‘effect’ reason (karya-hetu ), a
natural reason (svabhava-hetu ), or a reason based on nonobservation ( anupalabdhi -
hetu). 39
In each case, the presence of the reason in some sense necessitates the presence
of the target. An effect-reason is a property whose presence is causally necessitated
by the presence of the target property - for example, inferring that the mountain
has fire on it, because of smoke above it. The reason-target relation is a causal
relation. Clearly one can, and later philosophers 40 indeed did, extend this to
cover other species of causal inference, such as cases when reason and target are
both effects of a common cause. The generalisation ‘night follows day’ is true,
38 For further discussion, see B. K. Matilal, ’’Introducing Indian Logic”, in Matilal (1998),
reprinted in Jonardon Ganeri ed., ( Indian Logic: A Reader
39 Dharmaklrti, Nydyabindu II 11-12.
40 See Moksakaragupta’s eleventh century Tarkabhasa or Language of Reason. Yuichi Ka-
jiyama, An Introduction to Buddhist Philosophy: An Annotated Translation of the Tarkabhasa
of Moksakaragupta, Memoirs of the Faculty of Letters (Kyoto) 10 (1966), pp. 74-76.
Indian Logic
351
not because day causes night but because both day and night are caused by the
rotation of the earth. An example often cited is the inference of lemon-colour from
lemon-taste, when both are products of the same cause, viz. the lemon itself. Still
another example is the inference of ashes from smoke: ashes and smoke are both
effects of fire. Such an inference has two steps. First, fire is inferred from smoke;
second, ash is inferred from fire. The second step, in which we infer an effect from
its cause, is possible only because ash is a necessary effect of fire.
A natural reason is one whose presence metaphysically necessitates that of the
target property, for example the inference that something is a tree because it is a
simsapa (a species of tree). Dharmaklrti appears to regard the law “all sirnsapas
are trees” as necessarily true, even if its truth has to be discovered by observation,
and thus to anticipate the idea that there are a posteriori necessities. 41 He states,
surprisingly, that the reason-target relation in such inferences is the relation of
identity. Why? Perhaps his idea is that the two properties being-a-simsapa and
being-a-tree are token-identical, for the particular tree does not have two distinct
properties, being-a-simsapa and a separate property being-a-tree, any more than
something which weighs one kilogramme has two properties, having-weight and
having-weight-one-kilogramme. The properties as types are distinct, but their
tokens in individual objects are identical. Trope-theoretically, the point can easily
be understood. The very same trope is a member of two properties, one wider in
extension than the other, just as the class of blue tropes is a subset of the class of
colour tropes. But a blue object does not have two tropes - one from the class of
blue tropes and one from the class of colour tropes. It is the self-same trope.
Is absence of evidence evidence of absence? According to Dharmaklrti, nonob¬
servation sometimes proves absence: my failure to see an object, when all the con¬
ditions for its perception are met, is grounds for an inference that it is not here.
The pattern of argument such inferences exemplify was known to the medievals as
argumentum ad ignorantiam, or an ‘argument from ignorance.’ The pattern occurs
whenever one infers that p on the grounds that there is no evidence that p is false.
Dharmaklrti states that the argument depends on the object’s being perceptible,
i.e. that all the conditions for its perception (other than its actual presence) are
met in the given situation. Douglas Walton, in a major study of arguments from
ignorance, 42 claims that they depend for their validity on an implicit conditional
premise — if p were false, p would be known to be false. The characteristics of an
argument from ignorance are then a ‘lack-of-knowledge’ premise — it is not known
that not-p, and a ‘search’ premise — if p were false, it would be known that not-p.
The underlying hidden premise mentioned by Dharmaklrti seems to be exactly the
one Walton gives: if the object were here, one would see it. The necessity here is
subjunctive. The argument has a presumptive status - one has a right to presume
the conclusion to be true to the extent that one has searched for and failed to find
counter-evidence. It is this idea that is strikingly absent in Dinnaga. Warranted
extrapolation depends not on the mere nonobservation of counterexamples, but
41 Pramanavarttika I, 39-42.
42 Walton (1996).
352
Jonardon Ganeri
on one’s failing to find them in the course of a suitably extensive search.
In each of the three cases, the universal relation between reason and target is
a relation not of coincidence but of necessity - causal, metaphysical or subjunc¬
tive. DharmakTrti’s solution to the problem of induction, then, is to claim that
observation supports a generalisation only when that generalisation is lawlike or
necessary. In this, I think he anticipates the idea that the distinction between
lawlike and accidental generalisations is that only the former support the coun-
terfactual ‘if the reason property were instantiated here, so would be the target
property’. In such a context, let us note, the observation of even a single positive
example might sometimes be sufficient to warrant the extrapolation: I infer that
any mango is sweet having tasted a single mango; I infer that any fire will burn
having once been burnt.
Extrapolation is warranted when the reason-target is lawlike, but it does not
follow that the extrapolator must know that it is lawlike. What Dharmaklrti
has succeeded in doing is to describe the conditions under which extrapolation
works — the conditions under which one’s actions, were they to be in accordance
with the extrapolation, would meet with success. It is a description of the type
of circumstance in which extrapolation is rewarded (i.e. true — if, as it seems,
Dharmaklrti has a pragmatic theory of truth 43 ). As to how, when or whether one
can know that one is in such a circumstance, that is another problem altogether
and not one that Dharmaklrti has necessarily to address. For a general theory of
rationality issues in conditions of the form ‘in circumstances C, it is rational to
do <f>' or ‘in circumstances C, it is rational to believe p\ And this is precisely the
form Dharmakirti’s conditions take.
2-4 The Jaina reformulation of the triple condition
Dinnaga had argued that there are three marks individually necessary and jointly
sufficient for the warranted extrapolation from reason to target (§2.2). They are
(1) that the reason be present in the site of the extrapolation, (2) that the reason
be present (only) in what is similar to the target, and (3) that the reason be absent
in what is dissimilar to the target. The second of these conditions is, arguably,
equivalent to the third, which asserts that the reason property is absent when
the target property is absent. That was supposed to capture the idea of a ‘no
counterexample’ condition, according to which an extrapolation is warranted just
in case there is nothing in which the reason is present but not the target. What
happens to this account if one allows, as the Jaina logicians do, that a property
and its absence be compossible in a single object? 44 What happens is that the
three marks cease to be sufficient for warranted extrapolation. In particular, the
third mark no longer captures the idea behind the ‘no counterexample’ condition.
43 Shoryu Katsura, “DharmakTrti’s Concept of Truth,” Journal of Indian Philosophy 12 (1984),
pp. 213-235. Georges B. J. Dreyfus, Recognizing Reality: Dharmakirti’s Philosophy and its
Tibetan Interpretations (Albany: State University of New York Press, 1997), chapter 17.
44 See B. K. Matilal, The Central Philosophy of Jainism (Ahmedabad: L. D. Institute of
Indology, 1981).
Indian Logic
353
For now the absence of the reason property in a place where the target is absent
does not preclude its presence there too! So the third mark can be satisfied and
yet there still be counterexamples — cases of the presence of the reason together
with the absence of the target.
The Jainas indeed claim that the three marks are neither necessary nor suffi¬
cient for warranted extrapolation. Their response is to substitute for the three
marks a new, single, mark. It is clear that if the presence and absence of a prop¬
erty are compossible, then a distinction needs to be drawn between absence and
nonpresence. The first is consistent with the presence of the property; the second
is not. Early post-Diiinaga Jainas like Akalaiika and Siddhasena described the
new mark in quasi-Buddhistic terms, as ‘no presence without’ ( a-vina-bhava ) —
i.e. no presence of the reason without the target. Thus Akalaiika: 45
An extrapolation is a cognition of what is signified from a sign known to
have the single mark of no presence without the target {sadhyavinabhava ).
Its result is blocking and other cognitions.
The relata of the causality and identity relations cannot be cognised
without the suppositional knowledge ( tarka) of their being impossible
otherwise, [which is] the proof that this is the single mark even without
those relations. Nor is a tree the own-nature ( svabhava ) or the effect
(karya) of such things as shade. And there is no disagreement here.
There is an obvious reference to and criticism of DharmakTrti here, 46 and also a
mention of the important idea, which we have already discussed, that presumptive
argumentation (tarka) is what gives us knowledge of the universal generalisations
grounding extrapolations. The crucial difference from the Buddhists is in the
meaning of ‘no presence’. For the Jainas, it has to stand for nonpresence and not
for absence. That led them to reformulate the reason-target relation as a relation
of necessitation. Siddhasena:
The mark of a reason is ‘being impossible otherwise’ ( anyathanupannatva ) [Nyayavatara
22 ].
Vadideva Sfiri gives the developed Jaina formulation:
A reason has a single mark, ‘determined as impossible otherwise’. It
does not have three marks, for fallacies are then still possible [Pramananaya-
tattalokalamkara 3.11-12].
The idea is that the reason cannot be present if the target is not. It is impossible
for the reason to be present otherwise than if the target is present. The presence
of the reason necessitates the presence of the target.
45 Lag hiy astray a, verse 12.
46 0n Akalanka on DharmakTrti: Nagin J. Shah, Akalanka’s Criticism of DharmakTrti’s Philos¬
ophy (Ahmedabad: L. D. Institute, 1967), pp. 267-270.
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Jonardon Ganeri
I said that Diiinaga’s three marks are, for the Jainas, neither necessary nor suf¬
ficient. They are not sufficient because they permit extrapolation when the reason
is both present and absent, and the target nonpresent. On what grounds are they
thought not to be necessary? The theory of extrapolation as developed first by the
early Naiyayikas and then by Dinnaga has a built-in simplifying assumption. The
assumption is that extrapolation is always a matter of inferring from the presence
of one property in an object to the presence of a second property in that same
object. But that assumption excludes many intuitively warranted extrapolations.
The main examples considered by the Jainas are: (i) the sakata star-group will
rise because krttika star-group has risen; (ii) the sun is above the horizon because
the earth is in light; (iii) there is a moon in the sky because there is a moon in the
water.
These examples are said to prove that the first of Dinnaga’s three marks, that
the reason property is present in the site, is not a necessary condition on warranted
extrapolation. And yet, while it is certainly desirable to broaden the reach of the
theory to cover new patterns of extrapolative inference, it is not very clear what
these examples show. What is the underlying generalisation? What are the similar
and dissimilar examples? In the first case, the extrapolation seems to be grounded
in the universal generalisation ‘whenever the krttika arises, so too does the sakata .’
But then there is indeed a single site of extrapolation — the present time. The
inference is: the sakata will rise now because krttika has now risen. A similar point
could be made about the second example. There seems indeed to be an implicit
temporal reference in both of the first two cases, an extrapolation grounded in a
universal generalisation over times.
The third case is more convincing, yet here too one might try to discern a com¬
mon site. For the true form of the extrapolation is: the moon is in the sky because
it is reflected in the water, an extrapolation grounded in a universal generalisation
of the form ‘objects cause their own reflections’. Certainly, however, there are
patterns of extrapolation for which the ‘single site’ condition does not hold. If, for
example, one can find a universal generalisation of the form l Vx3y(Fx -4- Gy)\
then from ‘3 xFx' one can infer ‘3xGx’. Perhaps this is the pattern of extrap¬
olation the Jainas intend to exemplify with their example of a sky-moon and a
water-moon. If so, it is represents an important criticism of a simplifying, but in
the end also restricting, assumption in the classical theory of extrapolation.
3 JAINA CONTRIBUTIONS IN INDIAN LOGIC: THE LOGIC OF
ASSERTION
3.1 Rationality and Consistency
What is the rational response when confronted with a set of propositions each of
which we have some reason to accept, and yet which taken together form an incon¬
sistent class? This was, in a nutshell, the problem addressed by the Jaina logicians
of classical India, and the solution they gave is, I think, of great interest, both for
Indian Logic
355
what it tells us about the relationship between rationality and consistency, and for
what we can learn about the logical basis of philosophical pluralism. The Jainas
claim that we can continue to reason in spite of the presence of inconsistencies,
and indeed construct a many-valued logical system tailored to the purpose. My
aim in this chapter is to offer an interpretation of that system and to try to draw
out some of its philosophical implications.
There was in classical India a great deal of philosophical activity. Over the years,
certain questions came to be seen as fundamental, and were hotly contested. Are
there universals? Do objects endure or perdure? Are there souls, and, if so, are
they eternal or non-eternal entities? Do there exist wholes over and above collec¬
tions of parts? Different groups of philosophers offered different answers to these
and many other such questions, and each, moreover, was able to supply plausible
arguments in favour of their position, or to offer a world-view from which their
particular answers seemed true. The body of philosophical discourse collectively
contained therefore, a mass of assertions and contradictory counter-assertions, be¬
hind each of which there lay a battery of plausible arguments. Such a situation
is by no means unique to philosophical discourse. Consider, for instance, the cur¬
rent status of physical theory, which comprises two sub-theories, relativity and
quantum mechanics, each of which is extremely well supported, and yet which
are mutually inconsistent. The same problem is met with in computer science,
where a central notion, that of putting a query to a data-base, runs into trou¬
ble when the data-base contains data which is inconsistent because it is coming
in from many different sources. For another example of the general phenomenon
under discussion, consider the situation faced by an investigator using multiple-
choice questionnaires, when the answers supplied in one context are in conflict
with those supplied in another. Has the interrogee said ‘yes’ or ‘no’ to a given
question, when they said ‘yes’ under one set of conditions but ‘no’ under another?
Do their answers have any value at all, or should we simply discard the whole
lot on account of its inconsistency? Perhaps the most apposite example of all is
the case of a jury being presented with the evidence from a series of witnesses.
Each witness, we might suppose, tells a consistent story, but the total evidence
presented to the jury might itself well be inconsistent.
The situation the Jainas have in mind is one in which a globally inconsistent
set of propositions, the totality of philosophical discourse, is divided into sub¬
sets, each of which is internally consistent. Any proposition might be supported
by others from within the same subset. At the same time, the negation of that
proposition might occur in a distinct, though possibly overlapping subset, and
be supported by other propositions within it. Each such consistent sub-set of a
globally inconsistent discourse, is what the Jainas call a “standpoint” ( naya ). A
standpoint corresponds to a particular philosophical perspective.
Let us say that a proposition is arguable if it is assertible within some standpoint,
i.e. if it is a member of a mutually supporting consistent set of propositions.
The original problem posed was this: what is the rational reaction to a class
of propositions, each of which is, in this sense, arguable, yet which is globally
356
Jonardon Ganeri
inconsistent? It seems that there are three broad types of response. The first,
which I will dub doctrinalism, is to say that it will always be possible, in principle,
to discover which of two inconsistent propositions is true, and which is false. Hence
our reaction should be to reduce the inconsistent set to a consistent subset, by
rejecting propositions which, on close examination, we find to be unwarranted.
This is, of course, the ideal in philosophical debate, but it is a situation we are
rarely if ever in. The problem was stipulated to be one such that we cannot decide,
as impartial observers, which of the available standpoints, if any, is correct. If
doctrinalism were the only option, then we would have no choice but to come
down in favour of one or other of the standpoints, basing our selection, perhaps
on historical, cultural, or sociological considerations, but not on logical ones.
A second response is that of scepticism. Here the idea is that the existence both
of a reason to assert and a reason to reject a proposition itself constitutes a reason
to deny that we can justifiably either assert or deny the proposition. A justification
of a proposition can be defeated by an equally plausible justification of its negation.
This sceptical reaction is at the same time a natural and philosophically interesting
one, and indeed has been adopted by some philosophers, notably Nagarjuna in
India and the Pyrrhonic sceptics as reported by Sextus Empiricus. Sextus, indeed
states as the first of five arguments for scepticism, that philosophers have never
been able to agree with one another, not even about the criteria we should use to
settle controversies.
The third response is that of pluralism, and this is the response favoured by the
Jainas. The pluralist finds some way conditionally to assent to each of the propo¬
sitions, and she does so by recognising that the justification of a proposition is
internal to a standpoint. In this way, the Jainas try “to establish a rapprochement
between seemingly disagreeing philosophical schools” 47 , thereby avoiding the dog¬
matism or “one-sidedness” from which such disagreements flow. Hence another
name for their theory was anekantavada, the doctrine'of “non-one-sidedness” , 48
In spite of appearances to the contrary, the sceptic and the pluralist have much
in common. For although the sceptic rejects all the propositions while the pluralist
endorses all of them, they both deny that we can solve the problem by privileging
just one position, i.e. by adopting the position of the doctrinalist. (It seems,
indeed, that scepticism and pluralism developed in tandem in India, both as critical
reactions to the system-based philosophical institutions.) Note too that both are
under pressure to revise classical logic. For the sceptic, the problem is with the
law of excluded middle, the principle that for all p, either p or ->p. The reason
this is a problem for the sceptic is that she wishes to reject each proposition p
without being forced to assent to its negation -<p. The pluralist, on the other
hand, has trouble with a different classical law, the law of non-contradiction, that
for all p, it is not the case both that p and that ->p, for she wishes to assent both
47 B.K. Matilal, The Central Philosophy of Jainism, Calcutta University Press, Calcutta
1977:61.
48 For a good outline of these aspects of Jaina philosophical theory, see B.K. Matilal, The
Central Philosophy of Jainism , and P. Dundas, The Jains , Routledge Press, London 1992.
Indian Logic
357
to the proposition p and to its negation. While a comparative study of the two
responses, sceptical and pluralist, would be of interest, I will here confine myself
to developing the version of pluralism developed by the Jainas, and discussing the
extent to which their system becomes paraconsistent. It is very often claimed that
the Jainas ‘embrace’ inconsistency, but I will be arguing that this is not so, that
we can understand their system by giving it a less strongly paraconsistent reading.
3.2 Jaina seven-valued logic
The Jaina philosophers support their pluralism by constructing a logic in which
there are seven distinct semantic predicates ( bhahgi ), which, since they attach to
sentences, we might think of as truth-values (for a slightly different interpretation,
see Ganeri 2001, chapter 5). I will first set out the system following the mode of
description employed by the Jainas themselves, before attempting to reconstruct
it in a modern idiom. I will follow here the twelfth century author Vadideva Suri
(1086-1169 A.D.), but similar descriptions are given by many others, including
Prabhacandra, Mallisena and Samantabhadra. This is what Vadideva Suri says
( Pramana-naya-tattvalokalahkarah , chapter 4, verses 15-21): 49
The seven predicate theory consists in the use of seven claims about
sentences, each preceded by “arguably” or “conditionally” ( syat ), [all]
concerning a single object and its particular properties, composed of
assertions and denials, either simultaneously or successively, and with¬
out contradiction. They are as follows:
1. Arguably, it (i.e. some object) exists ( syad asty eva). The first
predicate pertains to an assertion.
2. Arguably, it does not exist (syan nasty eva). The second predicate
pertains to a denial.
3. Arguably, it exists; arguably, it doesn’t exist ( syad asty eva syan
nasty eva). The third predicate pertains to successive assertion
and denial.
4. Arguably, it is ‘non-assertible’ (syad avaktavyam eva). The fourth
predicate pertains to a simultaneous assertion and denial.
5. Arguably, it exists; arguably it is non-assertible ( syad asty eva
syad avaktavyam eva). The fifth predicate pertains to an assertion
and a simultaneous assertion and denial.
6. Arguably, it doesn’t exist; arguably it is non-assertible ( syan nasty
eva syad avaktavyam eva). The sixth predicate pertains to a de¬
nial and a simultaneous assertion and denial.
49 Vadideva Suri: 1967, Pramana-naya-tattvalokalamkara, ed. and transl. H. S. Battacharya,
Jain Sahitya Vikas Mandal, Bombay.
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Jonardon Ganeri
7. Arguably, it exists; arguably it doesn’t exist; arguably it is non-
assertible ( syad asty eva syan nasty eva syad avaktavyam eva).
The seventh predicate pertains to a successive assertion and denial
and a simultaneous assertion and denial.
The structure here is simple enough. There are three basic truth-values, true (t),
false (f), and non-assertible (u). There is also some means of combining basic
truth-values, to form four further compound values, which we can designate tf,
tu, fu and tfu. There is a hint too that the third basic value is itself somehow a
product of the first two, although by some other means of combination - hence
the talk of simultaneous and successive assertion and denial. Thus, in Jaina seven
valued logic, all the truth-values are thought to be combinations in some way or
another of the two classical values.
There is, however, a clear risk that the seven values in this system will collapse
trivially into three. For if the fifth value, tu, means simply “true and true-and-
false”, how is it distinct from the fourth value, u, “true-and-false”? No recon¬
struction of the Jaina system can be correct if it does not show how each of the
seven values is distinct. The way forward is to pay due attention to the role of
the conditionalising operator “arguably” ( syat). The literal meaning of “ syaF is
“perhaps it is”, the optative form of the verb “to be”. The Jaina logicians do not,
however, use it in quite its literal sense, which would imply that no assertion is
not made categorically, but only as a possibility-claims. Instead, they use it to
mean “from a certain standpoint” or “within a particular philosophical perspec¬
tive”. This is the Jaina pluralism: assertions are made categorically, but only from
within a particular framework of supporting assertions. If we let the symbol “V”
represent “syaf, then the Jaina logic is a logic of sentences of the form “Vp”, a
logic of conditionally justified assertions. As we will see, it resembles other logics
of assertion, especially the ones developed by Jaskowski 50 and Rescher 51
The first three of the seven predications now read as follows:
1. |p| = t iff Vp.
In other words, p is true iff it is arguable that p. We are to interpret this as saying
that there is some standpoint within which p is justifiably asserted. We can thus
write it as
1. |p| = t iff 3a a : p,
where “<r : p” means that p is arguable from the standpoint a. For the second
value we may similarly write,
2. |p| = / iff V
50 Jaskowski, S.: 1948, “Propositional calculus for contradictory deductive systems”; English
translation in Studia Logica 24 : 143 - 157 (1969).
51 Rescher, N.: 1968, Topics in Philosophical Logic , Reidel, Dordrecht.
Indian Logic
359
That is,
|p| = / iff a : ->p.
The third value is taken by those propositions whose status is controversial, in
the sense that they can be asserted from some standpoints but their negations
from others. These are the propositions which the Jainas are most concerned to
accommodate. Thus
3. \p\ = tf iff \p\ = t & \p\ = /.
I.e.
\p\ — tf iff Vp & V-ip,
or again
\p\ = tf iff 3a a : p & 3a a : ->p.
This way of introducing a new truth-value, by combining two others, may seem
a little odd. I think, however, that we can see the idea behind it if we approach
matters from another direction. Let us suppose that every standpoint is such that
for any given proposition, either the proposition or its negation is assertible from
within that standpoint. Later, I will argue that the Jainas did not want to make
this assumption, and that this is what lies behind their introduction of the new
truth-value “non-assertible”. But for the moment let us make the assumption,
which is tantamount to supposing that every standpoint is 'optimal’, in the sense
that for any arbitrary proposition, it either supplies grounds for accepting it, or
else grounds for denying it. There are no propositions about which an optimal
standpoint is simply indifferent. Now, with respect to the totality of actual optimal
standpoints, a proposition can be in just one of three states: either it is a member
of every optimal standpoint, or its negation is a member of every such standpoint,
or else it is a member of some, and its negation of the rest. If we number these
three states, 1, 2 and 3, and call the totality of all actual standpoints, £, then the
value of any proposition with respect to £ is either 1, 2 or 3. The values 1, 2 and
3 are in fact the values of a three-valued logic, which we can designate M3. There
is a correspondence between this logic and the system introduced by the Jainas
(J3, say). The idea, roughly is that a proposition has the value ‘true’ iff it either
has the value 1 or 3, it has the value ‘false’ iff it either has the value 2 or 3, and
it has the value ‘tf’ iff it has the value 3. Hence the three values introduced by
the Jainas represent, albeit indirectly, the three possible values a proposition may
take with respect to the totality of optimal standpoints.
Before elaborating this point further, we must find an interpretation for the
Jainas’ fourth value “non-assertible”. Bharucha and Kamat offer the following
analysis of the fourth value:
The fourth predication consists of affirmative and negative statements
made simultaneously. Since an object X is incapable of being ex¬
pressed in terms of existence and non-existence at the same time, even
allowing for Syad, it is termed ‘indescribable’. Hence we assign to the
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fourth predication ... the indeterminate truth-value I and denote the
statement corresponding to the fourth predication as (p&->/>). 52
Bharucha and Kamat’s interpretation is equivalent to
4. |p| = u iff V(p&-ip),
that is
\p\ = u iff 3a a : (p&->p).
Thus, for Bharucha and Kamat, the Jaina system is paraconsistent because it
allows for standpoints in which contradictions are justifiably assertible. This seems
to me to identify the paraconsistent element in the Jaina theory in quite the
wrong place. For while there may be certain sentences, such as the Liar, which
can justifiably be both asserted and denied, this cannot be the case for the wide
variety of sentences which the Jainas have in mind, sentences like “There exist
universal” and so on. Even aside from such worries, the current proposal has
a technical defect. For what now is the fifth truth-value, tu? If Bharucha and
Kamat are right then it means that there is some standpoint from which l p’ can
be asserted, and some from which can be asserted. But this is logically
equivalent to u itself. The Bharucha and Kamat formulation fails to show how we
get to a seven-valued logic.
Another proposed interpretation is due to Matilal. Taking at face-value the
Jainas’ elaboration of the fourth value as meaning “simultaneously both true and
false”, he says
the direct and unequivocal challenge to the notion of contradiction in
standard logic comes when it is claimed that the same proposition is
both true and false at the same time in the same sense. This is exactly
accomplished by the introduction of the [fourth] value - “Inexpressible”,
which can also be rendered as paradoxical. 53
Matilal’s intended interpretation seems thus to be
4. |p| = u iff V(p,->p),
i.e. |p| = u iff 3a(a : pfoa : -> p).
Matilal’s interpretation is a little weaker than Bharucha and Kamat, for he does
not explicitly state that the conjunction ‘p&-p’ is asserted, only that both con-
juncts are. Admittedly, the difference between Matilal and Bharucha and Kamat
is very slight, and indeed only exists if we can somehow make out the claim that
52 Bharucha, F. and Kamat, R. V.: 1984, “Syadvada theory of Jainism in terms of deviant
logic”, Indian Philosophical Quarterly, 9: 181 - 187; 183.
53 Matilal, B. K.: 1991, “Anekanta: both yes and no?”, Journal of Indian Council of Philo¬
sophical Research, 8: 1 - 12; 10.
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361
both a proposition and its negation are assertible without it being the case that
their conjunction is. For example, we might think that the standpoint of physical
theory can be consistently extended by including the assertion that gods exists,
and also by including the assertion that gods do not exist. It would not follow that
one could from any standpoint assert the conjunction of these claims. Yet whether
there is such a difference between Matilal’s position and that of Bharucha and Ka-
mat is rather immaterial, since Matilal’s proposal clearly suffers from the precisely
the same technical defect as theirs, namely the lack of distinctness between the
fourth and fifth values.
Tere is another interpretation, one which gives an intuitive sense to the truth-
value “non-assertible”, sustains the distinctness of each of the seven values, but
does not require us to abandon the assumption that standpoints are internally
consistent. Recall that we earlier introduced the idea of an optimal standpoint,
by means of the assumption that for every proposition, either it or its negation
is justifiably assertible from within the standpoint. Suppose we now retract that
assumption, and allow for the existence of standpoints which are just neutral about
the truth or falsity of some propositions. We can then introduce a new value as
follows:
4. |p| = u <=> 3ct(->(< 7 : p)&-i(cr : ->p)).
Neither the proposition nor its negation is assertible from the standpoint. For
example, neither the proposition that happiness is a virtue nor its negation receives
any justification from the standpoint of physical theory. We have, in effect, rejected
a commutativity rule, that if it not the case that ‘p’ is assertible from a standpoint
a then ‘-ip’ is assertible from a and vice versa [->(<r : p) (a : -ip)]. Our
new truth-value, u, is quite naturally called “non-assertible”, and it is clear that
the fifth value, tu, the conjunction of t with u, is not equivalent simply with u.
The degree to which the Jaina system is paraconsistent is, on this interpretation,
restricted to the sense in which a proposition can be tf, i.e. both true and false
because assertible from one standpoint but deniable from another. It does not
follow that there are standpoints from which contradictions can be asserted.
Why have so many writers on Jaina logic have felt that Jaina logic is paracon¬
sistent in the much stronger sense. The reason for this belief is the account which
some of the Jainas themselves give of the meaning of their third basic truth-value,
“non-assertible”. As we saw in the passage from Vadideva Suri, some of them say
that a proposition is non-assertible iff it is arguably both true and false simulta¬
neously, as distinct from the truth value tf, which is successively arguably true
and arguably false. We are interpreting the Jaina distinction between successive
and simultaneous combination of truth-values in terms of a scope distinction with
the operator “arguably”. One reads “arguably (f&/)”, the other “(arguably t)
& (arguably /)”. If this were the correct analysis of the fourth truth-value, then
Jaina logic would indeed be strongly paraconsistent, for it would be committed
to the assumption that there are philosophical positions in which contradictions
are rationally assertible. Yet while such an interpretation is, on the face of it,
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the most natural way of reading Vadideva Suri’s elaboration of the distinction
between the third and fourth values, it if far from clear that the Jaina pluralism
really commits them to paraconsistency in this strong form. Their goal is, to
be sure, to reconcile or synthesise mutually opposing philosophical positions, but
they have no reason to suppose that a single philosophical standpoint can itself be
inconsistent. Internal consistency was, in classical India, the essential attribute of
a philosophical theory, and a universally acknowledged way to undermine the po¬
sition of one’s philosophical opponent was to show that their theory contradicted
itself. The Jainas were as sensitive as anyone else to allegations that they were
inconsistent, and strenuously denied such allegations when made. I have shown
that it is possible to reconstruct Jaina seven-valued logic in a way which does not
commit them to a strongly paraconsistent position.
The interpretation I give to the value “non-assertible” is quite intuitive, al¬
though it does not mean “both true and false simultaneously'. My interpre¬
tation, moreover, is supported by at least one Jaina logician, Prabhacandra.
Prabhacandra, who belongs to the first part of the ninth century C.E., is one
of the few Jainas directly to address the question of why there should be just
seven values. What he has to say is very interesting:
(Opponent:) Just as the values ‘true’ and ‘false’, taken successively,
form a new truth-value ‘true-false’, so do the values ‘true’ and ‘true-
false’. Therefore, the claim that there are seven truth-values is wrong.
(Reply:) No: the successive combination of ‘true’ and ‘true-false’ does
not form a new truth-value, because it is impossible to have ‘true’
twice. ... In the same way, the successive combination of ‘false’ and
‘true-false’ does not form a new truth-value.
(Opponent:) How then does the combination of the first and the fourth,
or the second and the fourth, or the third and the fourth, form a new
value?
(Reply:) It is because, in the fourth value “non-assertible”, there is
no grasp of truth or falsity. In fact, the word “non-assertible” does
not denote the simultaneous combination of truth and falsity. What
then? What is meant by the truth-value “non-assertible” is that it is
impossible to say which of ‘true’ and ‘false’ it is. 54
This passage seems to support the interpretation offered above. When talking
about the “law of non-contradiction” in a deductive system, we must distinguish
between two quite different theses: (a) the thesis that K -i(p&:->p)” is a theorem in
the system, and (b) the thesis that it is not the case that both ‘p’ and ‘-p’ are
theorems. The Jainas are committed to the first of these theses, but reject the
second. This is the sense in which it is correct to say that the Jainas reject the
“law of non-contradiction”.
54 Prabhacandra: 1941, Prameyakamalamartanda, ed. M. K. Shastri, Nirnayasagar Press,
Bombay; p. 683 line 7 ff.
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363
I showed earlier that when we restrict ourselves to optimal standpoints, the total
discourse falls into just one of three possible states with respect to each system.
The Jainas have a set/en-valued logic because, if we allow for the existence of
non-optimal standpoints, standpoints which are just neutral with respect to some
propositions, then, for each proposition, p say, the total discourse has exactly seven
possible states. They are as follows:
1. p is a member of every standpoint in E.
2. -ip is a member of every standpoint in E.
3. p is a member of some standpoints, and -<p is a member of the rest.
4. p is a member of some standpoints, the rest being neutral.
5. is a member of some standpoints, the rest being neutral.
6. p is neutral with respect to every standpoint.
7. p is a member of some standpoints, ->p is a member of some other stand¬
points, and the rest are neutral.
Although Jainas do not define the states in this way, but rather via the possible
combinations of the three primitive values, t, f and u, it is not difficult to see that
the two sets map onto one another, just as they did before. Thus t = (1, 3, 4, 7),
/ = (2, 3 5, 7), tf = (3, 7), and so on.
Using many-valued logics in this way, it should be noted, does not involve
any radical departure from classical logic. The Jainas stress their commitment
to bivalence, when they try to show, as Vadideva Suri did above, that the seven
values in their system are all products of combining two basic values. This reflects,
I think, a commitment to bivalence concerning the truth-values of propositions
themselves. The underlying logic within each standpoint is classical, and it is
further assumed that each standpoint or participant is internally consistent. The
sometimes-made suggestion 55 that sense can be made of many-valued logics if we
interpret the assignment of non-classical values to propositions via the assignment
of classical values to related items is reflected here in the fact that the truth-value
of any proposition p (i.e. |p|) has two values, the status of p with respect to
standpoint a (‘|p|o-’) derivatively has three values, and the status of p with respect
to a discourse E (‘|p|s’), as we have just seen, has seven.
Consider again the earlier example of a jury faced with conflicting evidence from
a variety of witnesses. The Jainas wouldn’t here tell us ‘who dun it’, for they don’t
tell us the truth-value of any given proposition. What they give us is the means
to discover patterns in the evidence, and how to reason from them. For example,
if one proposition is agreed on by all the witnesses, and another is agreed on by
some but not others, use of the Jaina system will assign different values to the two
55 Haack, S.: 1974, Deviant Logic , Cambridge University Press, Cambridge; 64.
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propositions. The Jainas, as pluralists, do not try to judge which of the witnesses
is lying and which is telling the truth; their role is more like that of the court
recorder, to present the totality of evidence in a maximally perspicuous form, one
which still permits deduction from the totality of evidence.
So far so good. But there is another worry now, one which strikes at the very
idea of using a many-valued logic as the basis for a logic of discourse. For, when
we come to try and construct truth-tables for the logical constants in such a logic,
we discover that the logic is not truth-functional. That is to say, the truth-value of
a complex proposition such as ‘p&g’, is not a function solely of the truth-values of
the constituent propositions ‘p’ and ‘q\ To see this, and to begin to find a solution,
I shall need briefly to describe the work of the Polish logician, Jaskowski, who was
the founder of discursive logics in the West, and whose work, in motivation at
least, provides the nearest contemporary parallel to the Jain a theory.
3.3 Jaskowski and the Jainas
Philosophical discourse is globally inconsistent, since there are many propositions
to which some philosophers assent while others dissent. The Jainas therefore
develop a logic of assertions-made-from-within-a-particular-standpoint, and note
that an assertion can be both arguably true, i.e. justified by being a member
of a consistent philosophical position, and at the same time be arguably false, if
its negation is a member of some other consistent philosophical standpoint. This
move is quite similar to that of the founder of inconsistent logics, Jaskowski, who
developed a “discussive logic” in which a proposition is said to be ‘discussively
true’ iff it is asserted by some member of the discourse.
Jaskowski motivates his paper “Propositional Calculus for Contradictory De¬
ductive Systems” with two observations. The first is that
any vagueness of the term a can result in a contradiction of sentences,
because with reference to the same object X we may say that “X is
a” and also “X is not a”, according to the meanings of the term a
adopted for the moment,
the second is that
the evolution of the empirical sciences is marked by periods in which the
theorists are unable to explain the results of experiments by a homo¬
geneous and consistent theory, but use different hypotheses, which are
not always consistent with one another, to explain the various groups
of phenomena. 56
He then introduces an important distinction between two properties of deductive
systems. A deductive system is said to be contradictory if it includes pairs of
theorems A and ->A which contradict each other. It is over-complete, on the
56 Jaskowski, S.: 1948, “Propositional calculus for contradictory deductive systems”; English
translation in Studia Logica , 24 : 143 - 157 (1969); 144.
Indian Logic
365
other hand, if every well-formed formula is a theorem of the system. In classical
logic, these two properties are conflated; hence the slogan “anything follows from
a contradiction”. The problem to which Jaskowski addresses himself, therefore, is
that of constructing a non-classical system which is contradictory but not over¬
complete. In classical logic, given two contradictory theses A, -i A, we may deduce
first that A&->A, using the ^-introduction or Adjunction Rule, A, B —¥ A&B.
Then, since A&->A iff for any arbitrary A and B, and since B&6~>B —> B
from ^-elimination or Simplification, A&.B —t A, it follows that B. More clearly:
1. A, —iA
2. A&z-iA, from 1 by Adjunction.
3. A&-iA iff BSz-'B, for any arbitrary A and B.
4. B&i^B —t B , by Simplification.
5. A&->A —> B, from 3 and 4.
6. B, from 2 and 5 by Modus Ponens. ■
To get an inconsistent (contradictory but not over-complete) system, at least one
step in this sequence must be broken. In Jaskowski’s new system, ‘discursive logic’,
it is the Adjunction Rule which no longer holds. Jaskowski considers the system
in which many different participants makes assertions, each thereby contributing
information to a single discourse. The best example, perhaps, is one already given,
the evidence presented to a jury by witnesses at a trial. Jaskowski then introduces
the notion of discursive assertion, such that a sentence is discursively asserted if
it is asserted by one of the participants in the discourse, and he notes that the
operator “it is asserted by someone that...” is a modal operator for the semantics
of which it should be possible to use an existing modal logic. Thus
A is a theorem of D2 iff <C>A,
where D2 is Jaskowski’s two-valued discursive logic, and “0” is the operator
“someone asserts that...”. For some reason, Jaskowski chooses a strong modal
system, S5, to give the semantics of this operator, but this is surely a mistake.
The reason is that the S5 modal principle ‘A ^A’ does not seem to hold for a
discursive system, since there will be truths which no-one asserts. It would not be
difficult, however, to use a weaker modal system than S5, for example S2° or S3 0 ,
which lack the above principle, as the basis for D2. (The characteristic axiom of
S4°, ‘00A C’A’, does not seem to hold in a discursive system: it can be assert-
ible from some standpoint that there is another standpoint in which p is assertible
without there being such a standpoint). The point to note is that, in most modal
systems, the Adjunction Rule fails, since it does not follow that the conjunction
A&R is possible, even if A is possible and B is separately possible. And this too,
is what we would expect from the discursive operator, for one participant may
assert A, and another B, without there being anyone who asserts the conjunction.
Jaskowski therefore arrives at a system which is contradictory, since both A and
->A can be theses, but, because it is non-adjunctive, is not over-complete.
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3-4 The Logical Structure of the Jaina System
The parallels in motivation between Jaskowski’s discursive logic, and the Jaina
system are unmistakable. There is, however, an important difference, to which I
alluded earlier. Modal logics are not truth-functional; one cannot, for example,
deduce the truth-value of ‘0(^4&S)’ from the truth-values of ‘<>A’ and ‘0 B’. And
it seems for the same reason that a discursive logic cannot be truth-functional
either. Suppose, for example, that we have two propositions A, and B, both of
which are assertible from (possibly distinct) standpoints, and hence both true in
the Jaina system. What is the truth-value of A&zB'? It seems that this proposition
could be either true, false, or both.
It is possible to offer a defence of the Jaina position here. For simplicity, let us
restrict ourselves to the Jaina system with only optimal standpoints and just three
truth-values. If my suggested defence works here, its extension to the full Jaina
system J7, would not be especially problematic. Consider again the three-valued
logic, M3, whose values were defined as follows:
|p| = 1 iff Vcr a : p.
jpj = 2 iff Vu a : ->p
|p| = 3 iff 3er a : p & 3 ct a : ->p.
These correspond to the three possible states of a totality of optimal stand¬
points. When we try to construct the truth-table for conjunction in such a system,
we find that it is non-truth-functional. Thus, consider the truth-value of ‘p&g’,
when |p| = |g| = 3. Here, |p&g| might itself be 3, but it might also be 2. Thus,
the truth-value of the conjunction is not uniquely determined by those of its con-
juncts. What is uniquely determined, however, is that the truth-value belongs
to the class (2, 3). To proceed, we can appeal to an idea first introduced by N.
Rescher in his paper “Quasi-truth-functional systems of propositional logic” , 57 A
quasi-truth-functional logic is defined there as one in which “some connectives are
governed by many-valued functions of the truth-values of their variables”. The
entries in the truth-table of such a logic are typically not single truth-values but
sets of values. It is clear that the system set up just now is, in this, sense, quasi¬
truth-functional. Now, as Rescher himself points out, a quasi-truth-functional
logic will always be equivalent to a multi-valued strictly truth-functional system.
The idea, roughly, is that we can treat a class of truth-values as constituting a new
truth-value. Typically, if the quasi-truth-functional system has n truth-values, its
strictly truth-functional equivalent will have 2” - 1 values (Rescher notes that “in
the case of a three-valued (T, F, I) quasi-truth-functional system we would need
seven truth-values, to represent: T, F, I, (T, F), (T, I), (F, I), (T, F, I)” but argues
that there are special reasons entailing that for a two-valued quasi-truth-functional
system we need four rather than three values.). The seven-valued system which
results in this way from the three-valued logic sketched above has, in fact, been
57 Rescher, N.: 1962, “Quasi-truth-functional systems of propositional logic”, Journal of Sym-
bolic Logic , 27: 1 - 10.
Indian Logic
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studied notably by Moffat 58 . I will therefore call it M7. An initially tempting
idea is to identify the Jaina system J7 with M7. This, however, will only work if
the fourth value, u, is defined thus:
|p| = u iff Vcr cr : p V V<7 cr : ->p.
For then l tu' in the Jaina system will be identical with T’ in the Moffat system,
etc. This is, however, not an interpretation which receives any textual support.
Instead, let us observe that there is a close connection between M7 and the
restricted Jaina system, J3. For note that the value (1, 3) in M7 is such that
\p\ — (1) 3) iff |p| = 1 V \p\ = 3
iff V<7 a : p V (3a a : p & 3a a : ~<p)
iff 3a a : p.
Thus (1, 3) in M7 is just the value ‘true’ in J3. Similarly, (1, 2) in M7 is just
the value ‘false’ in J3. Thus, although J3 is not strictly truth-functional, its truth-
tables are embedded in those of the Moffat logic, M7.
It is presumably possible to find a quasi-truth-functional system whose truth-
tables embed those of J7, the full Jaina system, in an entirely analogous way.
Thus, although the loss of Adjunction means that the Jaina logic J7, is not truth-
functional, its truth-table is embedded in a suitable quasi-functional system. The
lack of truth-functionality is not, after all, a fatal flaw in the Jaina approach.
3.5 Axiomatisation of the Jaina System
We have shown that it is possible to use many-valued truth-tables to formalise the
Jaina system. This was, in effect, the approach of the Jaina logicians themselves.
Yet it would surely be much better to proceed by axiomatising the modal stand¬
point operator, V. Once again we look to Rescher 59 . His work on what he calls
“assertion logics” is an extension of the work of Jaskowski. Rescher introduces a
system Al, with the following axiomatic basis:
(Al) (3 p)a : p
(A2) (<7 : p & a : q) D a : (p & q)
(A3) -i(7 : (p & -i p)
(R) If p h q, then a : p\~ a : q
[Nonvacuousness]
[Conjunction]
[Consistency]
[Commitment]
Note that one effect of the rule (R) is to ensure that the notion captured is not
merely explicit assertion but ‘commitment to assert’, for (R) states that from a
standpoint one may assert anything entailed by another of the assertions. I be¬
lieve that the Jainas would accept each of the axioms (Al) to (A3). Bharucha and
58 Moffat, D. C. and Ritchie, G. D.: 1990, “Modal queries about partially-ordered plans”, J.
Expt. Theor. Artif. Intell., 2: 341 - 368. See also Priest, G.: 1984, “Hypercontradictions,”
Logique et Analyse , 107: 237-43.
59 Rescher, N.: 1968, Topics in Philosophical Logic , Reidel, Dordrecht, chapter xiv.
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Jonardon Ganeri
Kamat, it may be noted, would reject (A3), while Matilal, as I have represented
him, would reject (A2). I have already argued that these claims are mistaken. In
particular, with regard to (A2), although it is true that the Jainas reject Adjunc¬
tion, what this means is that assertions made from within different standpoints
cannot be conjoined, not that assertions made within the same standpoint cannot
be conjoined.
We now introduce the modal standpoint operator, V “arguably”, via the defi¬
nition:
Vp iff (3 «t)< 7 : p,
and add the axioms of S3 0 or some other suitable modal system.
Rescher defines some further systems by adding further axioms, none of which,
I think, the Jainas would accept. For example, he defines A2 by adding to A1
the axiom that anything asserted by everyone is true [(Vcrju : p D p]. There is no
reason to suppose the Jainas commit themselves to this. The system J3, however,
is distinguished by the new axiom (A4):
(A4) -i(3<r)(-K7 : p&-i(T : ->p) [Optimality]
Rescher too proposes a “three-valued approach” to assertion logic, via the notion
of ‘the truth status of the assertion p with respect to an assertor’, written ‘|p|er’,
and the definitions:
|p|cr = T iff a : p,
— F iff a : (—ip), and
= I iff ->(cr : p)&->(<7 : —>p),
and he shows that using the axioms of Al, we can derive a quasi-truth-functional
logic for this system. These are not quite the Jaina values, as introduced earlier,
for they do not quantify over standpoints or assertors. It is clear, however, that
the Jaina system is of the same type as a modalised Rescher assertion logic. Their
innovation is to introduce three truth-values via the definitions given before (|p|s =
t iff (3er)(er : p); |p| s = / iff (3 <t)(<t : ~>p); and jp| s = u iff (3cr)(-'(tr : p )&—■(a :
-i p)), where ‘\p\V stands for ‘the status of the assertion p with respect to the
total discourse £’). It is this attempt to take a many-valued approach to the
modalised, rather than the unmodalised, version of assertion logic which generates
the extra complexity of the Jaina system. I have already noted that, since the
axiom “p D Vp” is lacking, the modal structure of the system will be no stronger
than that of S3 0 . Yet in principle there seems no reason to think that the Jaina
system cannot in this way be given an axiomatic basis.
3.6 Pluralism, Syncretism, and the Many-faceted View of Reality
The Jainas avoid dogmatism and a one-sided view of the world simply by noting
that assertions are only justified in the background of certain presuppositions or
conditions. It is perfectly possible for an assertion to be justified given one set of
presuppositions, and for its negation to be justified given another different set. The
Indian Logic
369
Jainas’ ingenuity lies in the skill with which they developed a logic of discourse
to make more precise this natural idea. However, they also went beyond this,
for they added that every standpoint reveals a facet of reality, and that, to get
a full description of the world, what we need to do is to synthesise the various
standpoints. As Matilal puts it, “The Jainas contend that one should try to
understand the particular point of view of each disputing party if one wishes to
grasp completely the truth of the situation. The total truth ... may be derived
from the integration of all different viewpoints”. 60 But is this further step, the
step from pluralism to syncretism, a coherent step to take? In particular, how is
it possible to integrate inconsistent points of view? The point is made by Priest
and Routley, who, commenting on the Jaina theory, state that “...such a theory
risks trivialization unless some (cogent) restrictions are imposed on the parties
admitted as having obtained partial truth — restrictions of a type that might well
be applied to block amalgamations leading to violations of Non-Contradiction”. 61
Perhaps we can understand the Jaina position as follows. The so-called ‘inte¬
gration’ of two points of view, o\ and er 2 , does not mean the creation of some
new standpoint, which is the combination of the first two. For this would lead
to the formation of inconsistent standpoints unless implausible constraints were
placed on what can constitute a standpoint. Instead, what it means is that, if p is
assertible from some standpoint ay, then this fact, that p is assertible from U\ , can
itself be asserted from cr 2 and every other standpoint. In this way, each disputant
can recognise the element of truth in the other standpoints, by making explicit
the presuppositions or conditions under which any given assertion is made.
If correct, this idea has an interesting consequence. In moving from pluralism to
syncretism, the Jainas commit themselves to the claim that we are led to a complete
account of reality by integrating of all the different points of view . It follows from
this that every true proposition must be asserted within some standpoint, i.e.
“p D (3 a) (a : p) or “p D Vp”. Hence the move from pluralism to syncretism is a
move from a logic of assertibility based on S3 0 or weaker to one based on S3 or
stronger.
To conclude, we have seen how the Jainas developed a plausible and interesting
logic of philosophical discourse, how they did not (or need not) commit themselves
to the strongly paraconsistent position normally attributed to them, and how, as
they strengthened their position from one of pluralism to one of syncretism, they
had also to strengthen correspondingly the modal logic underlying the operator
“syaf.
60 Matilal, B. K.: 1977, The Central Philosophy of Jainism, Calcutta University Press, Cal¬
cutta.
61 Priest, G., Routley, R. Norman, .J. eds.: 1989, Paraconsistent Logic: Essays on the Incon¬
sistent, Philosophia Verlag, Munchen, p.17.
370
Jonardon Ganeri
4 LOGIC IN NAVYA-NYAYA: THE METAPHYSICAL BASIS OF LOGIC
4-1 The use of graphs in interpreting Vaisesika Ontology
Let us turn now to the Navya-Nyaya school, a school and a set of thinkers predis¬
posed towards the study of the metaphysical structure of the natural world, and
to the logical theory that is integral to this ontology. Three revisionary Nyaya
thinkers - Bhasarvajna (c. AD 950), Udayana (c. AD 1050), and Raghunatha
(c. AD 1500) - saw in effect that there is a graph-theoretic basis to the classical
Vaisesika notion of a category. I will show how the graph-theoretic interpretation
of their ideas lends itself to a distinctive treatment of negation, logical consequence
and number.
Classical Vaisesika lists six kinds of thing: substance, quality, motion, univer¬
sal, individuator, inherence. Later Vaisesika adds a seventh: absence. The basic
stuff of the cosmos in the Vaisesika world-view is atomic. Atoms are uncreatable,
indestructible, non-compound substances. Atoms can coalesce into composite sub¬
stances and can move. Indeed, the only changes in this cosmos are changes in the
arrangement, properties and positions of the atoms. Creation is a matter of coa¬
lescing, destruction of breaking (and even God does not create the cosmos ab nihilo,
but only ‘shapes’ it, as a potter shapes clay into a pot). A compound substance is
a whole, composed out of, and inhering simultaneously in each of, its parts. These
substances are individuated by the type and organisation of their parts. A ‘qual¬
ity’ in classical Vaisesika is a property-particular - for example, a particular shade
of blue colour or a distinct flavour (what one would now call a ‘thin’ property).
Qualities inhere in substances and in nothing other than substances. A ‘motion’
is another sort of particular; it too inheres in a substance and in nothing but a
substance. Universals inhere in substances, qualities and motions. A universal
inheres simultaneously in more than one, but has nothing inhering in it. Lastly,
the ‘individuator’ ( visesa ) is a distinctive and eponymous component in classical
Vaisesika ontology. An individuator inheres in and is unique to a particular atom:
it is that by which the atomic, partless substances are individuated. 62
Two principles lie at the heart of the Vaisesika system: a principle of identity
and a principle of change. The Vaisesika principle of change is this: a becomes
b iff the parts of a rearrange (perhaps with loss or gain) into the parts of b.
‘Motions’ are that in virtue of which the parts rearrange or stay together. There
are basic or partless parts, the atoms, which, precisely because they have no parts,
are incapable of becoming anything else. They move about but are eternal and
indestructible. The Vaisesika principle of identity is this: a — b iff the parts of a are
numerically identical to and in the same arrangement as the parts of b. ‘Qualities’
are that in virtue of which the parts are numerically identical or different. Atoms,
precisely because they are partless, require a different principle of identity: atoms
are distinct iff they have distinct individuators. Universals are limits on the degree
62 An excellent review of the details of Vaisesika ontology is Karl Potter ed., “Indian Meta¬
physics and Epistemology - The Tradition of Nyaya-Vaisesika up to Gangesa”, in The Encyclo¬
pedia of Indian Philosophies, vol. 2 (Delhi: Motilal Banarsidass, 1977), introduction.
Indian Logic
371
of possible difference and change. One thing cannot change into another thing of
an entirely different sort (a mouse into a mustard seed). One thing a can become
another thing b iff the same universal resides in both a and b, that is, if a and b
are of the same sort (as Udayana puts it, universals regulate causality).
This is the motivation for there being six ‘types’ of thing (substances, qualities,
motions, universals, individuators, inherence). The problem is to find a proper
philosophical basis for the notion of a ‘type’ of thing thus appealed to. In his
Laksanavali, Udayana reconstructs the categories in a new way, a way which I
shall claim explicates the notion of a type graph-theoretically. A graph is a simple
sort of algebraic structure, consisting of set of nodes or vertices, and a set of edges
(an edge being defined as a pair of nodes). A graph is ‘directed’ if the edges
have a direction. Graphs, like many other mathematical structures, are realised in
natural phenomena. A striking example is molecular structure: it is because the
structure of a molecule is a graph that one can use a graph to depict one:
H-O-H
The implicit structure of the Vaisesika ontology is that of a directed graph. The
inherence relation connects things in the ontology in inheror-inheree pairings. So
the substances, qualities, motions, universals and individuators are represented
as the nodes of a graph whose set of edges represent the inherence relation. A
fragment of the graph might look like this:
U
I
Q
f
t t
I I
This graph represents the following state of affairs: a universal U inheres in a
quality Q which inheres in a substance S. That substance is a dyad composed of
two atoms in which it inheres, and each of which has inhering in it an individuator
I. The structure of the world is a directed graph.
The nodes in a graph can be classified according to the number of edges termi¬
nating in them, and the number of edges starting from them: so the valency of
a node in a directed graph is an ordered pair of integers (n,m). What Udayana
saw in the Laksanavali is that things of different types in the Vaisesika ontol¬
ogy correspond to nodes of different valencies. His brilliant idea is to use the
idea of valency to define the categories of substance, quality, motion, universal,
and individuator. He begins with a classification of the categories into the four
valency-groups ( + ,+),( + , 0), (0,+) and (0,0): 63
63 Numbering of the verses in the Laksanavali follows Musashi Tachikawa, The Structure of
372
Jonardon Ganeri
5. Noneternal [= compound] substance, quality, motion, universal, and indi-
viduator inhere.
6. Eternal [i.e. atomic] substance, inherence, and absence lack the property of
inhering.
7. Substance, quality, and motion are inhered in.
8. Universal, individuator, inherence, and absence have nothing inhering in
them.
In particular then, atoms have valency (+, 0), universal and individuators have
valency (0,+), while compound substances, qualities and motions have valency
(+i +)■
Notice that Udayana says that the inherence relation itself has a valency —
(0,0). We should not take this to mean that the inherence relation is to be repre¬
sented by a node disconnected from the rest of the graph, but rather that it does
not correspond to any node in the graph at all. The first and most fundamental
graph-theoretic type distinction is the distinction between a node and an edge,
and the inherence relation is represented in a graph by the set of edges, not by
any node. The set of edges represents the extension of the inherence relation.
If the categories are to be distinguished from one another according to the va¬
lency of the nodes in that graph which is isomorphic to the world of things, then
further specification is needed. The distinction between universals and individu¬
ators is simple: an individuator has valency (0,1) while a universal has valency
(0,m), with m > 1:
202. A universal has nothing inhering in it, inheres, and is co-located with every
difference.
203. Individuators lack the property of being inhered in, inhere, and lack the
property of inhering by being co-located with every difference.
Udayana’s phrase ‘co-located with every difference’ is a technical device for
expressing the idea that a universal inheres in more than one. For if an inheror
inheres in exactly one thing x, then all other things are loci of difference-from-a;,
and the inheror is not co-located with difference-from-a:. However, if the inheror
inheres in two things x and y, then difference-from-a: is located in y and difference-
from-y is located in x, and the inheror is co-located with both differences. So
something co-located with every difference-from each of the things in which it
inheres is necessarily located in more than one thing. Notice that in classical
Vaisesika, individuators are said to have no universals inhering in them precisely
because they are fundamental units of individuation, having nothing in common
with one another.
Indian Logic
373
O
t
UNIVERSAL O ■*“ • —*■ O
I
o
INDIVIDUATOR.
I
o
Any node with valency (0, m) with m > 1 is now to be called a ‘universal’, and
any node with valency (0,1) is to be called an ‘individuator’:
The valency of atoms is different from that or qualities or motions, but we
still need a general definition of substance, covering both atomic and compound
substances. For compound substances, like universals but unlike atoms, inhere in
other things (their parts). Udayana in fact offers four definitions, of which the
first three repeat older definitions. The fourth definition, however, is completely
original:
9. A substance is not a substratum of absence of quality.
10. Or, it belongs to such a kind as inheres in what is incorporeal, inheres in
what is not incorporeal and does not inhere in what inheres in what is not
corporeal.
11. Or, it belongs to such a kind as inheres in space and in a lotus but not in
smell.
the World in Udayana’s Realism: A Study of the Laksandvali and the Kiranavali (Dordrecht:
Reidel Publishing Company, 1981).
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Jonardon Ganeri
12. Or, it is that in which inheres that in which inheres that which inheres.
The first of these definitions is the classical one in Vaisesika 64 — a substance
is that which possesses qualities. Udayana returns to this definition in his famous
but conservative commentary, the Kiran avail. He thinks of replacing it in the
more experimental LaksanavalT with a definition that makes no reference to any
other category and indeed is phrased entirely in terms of the notion of inherence:
a substance is ‘that in which inheres that in which inheres that which inheres’. In
other words, a substance is to be represented by a node like this:
o
I
o
I
SUBSTANCE 9
The point of the definition is that a substance possesses qualities, and qualities
possess universal, and nothing else in the ontology possesses something which pos¬
sesses something. For universals and individuators possess nothing, while qualities
and motions possess universals and nothing else.
Let us define a ‘path’ between one node and another in the obvious way: there
is a path from node x to node w if there is a sequence of nodes {x, y, ..., w} such
that there is an edge from x to y, an edge from y to z, ... , an edge from v to w . 65
Define the ‘length’ of a path as the number of edges between the first and the last
node. Udayana’s definition of a substance is now: a node is a substance iff there
is a path at least of length 2 leading to it. Substances inhere in their parts; so the
definition entails that every part of a substance is a substance.
The classical conception of qualities and motions makes them almost identical:
they both inhere only in substances, and they both are inhered in only by univer¬
sals. 66 Prasastapada’s remark 67 that the qualities other than contact, breaking,
number and separateness ‘inhere in one thing at a time’ should not be construed
as implying that they inhere in only one thing, but only that this group of qualities
are monadic (non-relational) properties. These features are enough to distinguish
qualities and motions from all else: from universals and individuators (which do
64 Vaisesikasutra 1.1.14: “The characteristic of a substance is to possess actions, qualities and
to be [their] inherence cause.”
65 In what follows, bold roman letters denote nodes in the graph, and italic letters denote the
entities those nodes represent.
66 Vaisesikasutra 1.1.15—6. Padarthadharmasamgraha 18. Section numbering in the
Padarthadharmasamgraha follows Karl Potter ed., “Indian Metaphysics and Epistemology -
The Tradition of Nyaya-Vaisesika up to Gangesa”, in The Encyclopedia of Indian Philosophies,
vol. 2 (Delhi: Motilal Banarsidass, 1977), pp. 282-303.
67 Padarthadharmasamgraha 50-51.
Indian Logic
375
not have anything inhering in them), and from substances (which are inhered in
by things that are themselves inhered in). It explains too why qualities cannot
inhere in qualities — if they did then they would be equivalent graph-theoretically
to substances.
QUALITY or MOTION
0 — 0 — 0
1
I
o
What is difficult is to find any principled way to distinguish between qualities
and motions. There was indeed a persistent revisionary pressure to assimilate these
two categories. Bhasarvajna 68 heads the revisionary move, stating unequivocally
that motions should be treated as qualities because, like qualities, they reside
in substances and possess universals. From a graph-theoretic perspective, this
revision is well motivated: qualities and motions are represented by nodes of the
same valency, and so are things of the same type. Udayana chooses the harder
way, and tries to formulate definitions that will accommodate the distinction. The
classical Vaisesika idea 69 that motions are what cause substances to come into
contact with one another is reflected in his definitions:
126. A quality belongs to such a kind as inheres in both contact and non-contact,
and does not inhere in the non-inherent cause of that sort of contact which
does not result from contact.
190. A motion belongs to such a kind as inheres in the non-inherent cause of
contact and does not inhere in contact.
These definitions introduce two new relations, contact and causation, neither
of which are explicable in terms of inherence nor belongs to the graph-theoretic
interpretation of the categories. The very success of that interpretation gives a
rationale to the revisionary pressure. Finding a pattern into which all but a few
items of some phenomenon fit grounds a presumption that those items are in some
way discrepant. This is a general principle of scientific and rational inquiry, and we
can see it been used by Bhasarvajna to motivate revisions in the classical Vaisesika
theory. Rationality appears here in the form of principled revision. 70
68 Nyayabhusana, p. 158.
69 Vaisesikasutra 1.1.16.
70 For later comment on Bhasarvajna’s revision: Karl Potter and Sibajiban Bhattacharyya eds.
376
Jonardon Ganeri
Let us define a Vaisesika graph as a connected directed graph each of whose
nodes is a substance, quality, motion, universal or individuator, where:
A substance is a node terminating a directed path of length 2.
A quality or motion is a node v with valency (+,+), such that the
initial node of any edge terminating in v has zero invalency [i.e. such
that qualities are not substances].
A universal is a node with valency (0,n) with n > 1.
An individuator is a node with valency (0,1).
Let us say further that node x inheres in a node y iff xy is an edge in a Vaisesika
graph. Then we can easily prove some results well-known to the Nyaya-Vaisesika
logicians:
LEMMA 1. No quality inheres in a quality.
Proof. The invalency of a quality x is nonzero, so any node in which x inheres
terminates a path of length 2. ■
LEMMA 2. Substances inhere only in substances.
Proof. Any node x in which a substance inheres terminates a path of length 2.
THEOREM 3. A Vaisesika graph has no directed cycles.
Proof. The elements of a directed cycle must have valency (+, +). So no universal
or individuator can be a member of such a cycle, because neither has nonzero
invalency. No quality or motion can be a member of a cycle, because only universals
and individuators inhere in qualities and motions, and there are no universals or
individuators in a cycle. No atomic substance can be a member of a cycle, because
atoms have zero outvalency. That leaves only cycles of compound substances. But
there can be cycles of substances only if a substance can have as a part something
of which it is a part and so (if the part-of relation is transitive) be a part of itself.
DEFINITION 4. The level of a node in a Vaisesika graph is the length of the
longest directed path leading to it. (Note — this is well-defined because there are
no directed cycles.)
THEOREM 5.
(i) All and only universals and individuators belong to level 0;
“Indian Philosophical Analysis — Nyaya-Vaisesika from Gangesa to Raghunatha Siromani”, in
The Encyclopedia of Indian Philosophies, vol. 6 (Delhi: Motilal Banarsidass, 1993), pp. 323,
525-528.
Indian Logic
377
(ii) All and only qualities and motions belong to level 1;
(Hi) All and only substances belong to levels 2 and below.
Proof, (i) Only universals and individuators have zero invalency, (ii) By Lemmas
1 and 2, qualities and motions are inhered in only by universals and individuators,
so belong to level 1. Substances do not belong to level 1 by definition, (iii). By
definition, any node in level 2 is a substance, and by Lemma 2, any node in level
n > 2 is a substance. ■
So the structure of the Vaisesika graph is like this:
0
universals, individuators
qualities, motions
substances all the way down
To what extent are we justified in adopting the graph-theoretical interpretation
of Navya-Nyaya? I propose the following Methodological Test: The graph-theoretic
interpretation is confirmed to the extent that it explains or predicts revisions made
to the classical Vaisesika system. Revisions include the introduction of a seventh
category absence, the assimilation of qualities and motions, the elimination of
individuators, the identification of co-extensive universals, the new account of
number. Let us say that a node x is redundant in G if its deletion, together with
the deletion of any edge incident to it, preserves all paths in G not containing x.
The resulting graph G* is a conservative contraction of G. Then we have, in effect,
the following revisions being proposed by Nyaya authors — (I) All individuators
are redundant in a Vaisesika graph (Raghunatha); (II) Two universals are co¬
extensive only if at least one is redundant (Udayana); and (III) Qualities and
motions are entities of the same type (Bhasarvajna).
4-2 Negation as absence
‘Absence’ in Navya-Nyaya is not the same as nonexistence. Fictional characters,
dream-objects and hallucinations are nonexistent: they do not exist as it were by
nature. It would be an absurdity to go in search of Hamlet in order to find out
378
Jonardon Ganeri
whether he really exists or not — his nonexistence is not a merely contingent lack
in the world of things. The absence of water on the moon, on the other hand, is a
contingent and concrete fact; so too is the absence of colour in my cheeks. Notice
the role of the phrases ‘of water’, ‘of colour’ here: an absence has an absentee —
that which the absence is an absence of. It also has a location (e.g. the moon, my
cheeks), and a time. So the proposal is to reparse the sentence L x does not occur
in y at time f’ as ‘an absence-of-rc occurs in y at t .’ For it is often the case that
the absence of something somewhere is more salient than any fact about what is
present there.
There is one relatively straightforward way to interpret the idea of absence
graph-theoretically. If x does not inhere in y, then there is no edge (x, y) in the
graph. Now for every graph, there is a dual. The dual has the same nodes as the
original graph, but has an edge between two nodes just in case the original does
not. So the dual graph does have an edge (x, y). Following this idea, one would
be led to say that absences are things of a different type to any presence because
they are edges in the dual graph, rather than edges or nodes in the original.
For various reasons, the Vaisesika do not consider this to be an adequate expla¬
nation of the category. One problem is that it makes absences more like relations
than ‘things’, and this does not keep to the spirit of the Vaisesika idea that ab¬
sences are entities. In fact, absences do display much relation-like behaviour
after all, absence is always the absence of x in y. Another objection, however, is
if absence is a new category, its introduction should result in an extension of the
original graph, and not in the introduction of a new graph, let alone a graph com¬
pletely disconnected from the original. For the connected world of things ought
not be represented by a pair of disconnected graphs. A third problem arises if we
admit something called ‘unpervaded’ occurrence, as we will see.
The Vaisesika idea is represent absences as nodes , related in new ways to the
nodes of the original graph. Here is how to do it. For each unconnected pair of
nodes (x, y), create a new node x' in the original graph. This new node will have
edges to x and to y, but they will be edges of two new types. The edge (x', x) is
an edge belonging to the extension of the absentee-absence (pratiyogita) relation,
which I shall signify as ‘=>.’ This represents the relation between an absence and
what the absence is of. The edge (x',y) is an edge belonging to the extension
of the ‘absential special relation’ ( abhavtya-svarupa-sambandha ), signified here by
This represents the occurrence relation between an absence and its location.
The relation between an absence and its location is clearly not the same as the
relation between a presence and its location (inherence, contact), for it is clear
that when a person is is absent from a room, their absence is not in the room in
the same sense that the other things in the room are.
These new nodes belong in a domain outside the system of levels, for they inhere
in nothing and nothing inheres in them (inherence, and the whole system of levels,
is a structure on presences). The modified graph is instead a concatenation of the
original graph of nodes and edges with a new structure of ‘absential nodes’ and
‘absential edges.’
Indian Logic
379
original graph modified original
O x O x
ft
ABSENCE O x '
I
Oy Oy
Navya-Nyaya theory of absence draws a type distinction between simple ab¬
sence (atyantabhava ) and difference (anyonyabhava ). Difference is the absence
of a relation of identity between two things. Here ‘x ^ y’ is paraphrased as ‘a
difference-from y occurs in x\ Graph-theoretically, the distinction between ab¬
sence and difference is a distinction between a negation on edges and a negation
on nodes in the original graph. For, trivially, every node is such that it is different
from every other node. One way to represent this would be to introduce a new
kind of ‘nonidentity’ edge into the graph, an edge which connects every node with
every other node. The Naiyayika, however, wants to the category of absence to
correspond to a domain of things rather than relations; so in the graph-theoretic
representation, differences have to be represented as nodes rather than edges. So
let us say that for every node x in the original graph, there is a new node x*.
Call it an ‘antinode’, x* is connected to every node in the graph. It is connected
to x by an edge of the absentee-absence type, and to every node other than x by
an absential location edge. There is a one-one correspondence between the new
domain of antinodes and the domain of original nodes.
O x
ft
DIFFERENCE
O «- O x * - o
I
o
The leading idea behind the graph-theoretic interpretation of the categories is
that a type of thing is a type of node, and node-types are determined by patterns
of possible valencies in the graph. It was for this reason that we did not need
earlier to the label the nodes. With the introduction of the category of absence,
380
Jonardon Ganeri
we have two higher-order type distinctions: the distinction between positive and
negative nodes, and the distinction among the negative nodes between absential
nodes and antinodes. Do these distinctions have a graph-theoretic explanation,
or must we allow ineliminable node-labels to demarcate presence nodes, absential
nodes and antinodes? What we do have now are three different types of edge
— corresponding to the relations of inherence, absence-absentee, and absential
location. So we might hope to distinguish between positive and negative nodes as
those which are not and those which are at the end of an absential edge. That is,
we make it a requirement that no positive node absentially qualify any other node.
Clearly, the suggestion will work only if the absence of an absence is not identical
to a presence. We will see in the next section that the graph-theoretically oriented
Raghunatha indeed denies that this is so. So as not to beg the question at this
point, and for the sake of pictorial clarity, I will continue to mark positive nodes
O and negative nodes Q differently.
What about the distinction between absential and antinodes? The traditional
way of making the distinction is to say that simple absence is the denial of inherence
(or some other nonidentity relation) and difference is the denial of identity. Graph-
theoretically, the distinctive feature of an antinode x* is that it absentially qualifies
every node other than x, while an absential node x' does not. Does this difference
fail when x is something which inheres in nothing (an atom, an individuator)?
No, because such things do not inhere in themselves — so x' unlike x* absentially
qualifies x. Indeed, this second contrast is itself sufficient to discriminate absential
nodes and antinodes.
The above treatment of absence is in effect a procedure for introducing new
nodes into the original graph. One set of new nodes fills the ‘gaps’ in that graph:
whenever there is no edge between two nodes, an absential node is introduced
between, and linked to, them. Another set of new nodes exactly mirrors the
original graph: for each node in the original, there is one and only one antinode,
linked to everything the original node is not. But now, having supplemented the
original graph with two sets of new nodes, nothing is to stop us from repeating
the procedure again — generating new sets of second-order absence nodes — and
to do this again and again. It seems that we have introduced a procedure for
the indefinite recursive expansion of the graph. Fortunately this does not in fact
happen. As we will now see, no subsequent recursion of the procedure after the
first produces any new nodes.
Prima facie, it seems plausible to reason as follows (as we will shortly see, this
reasoning turns out to be subject to an important caveat). If x is in y, then x',
the absence of x, is not in y, and so x", the absence of x', is in y. Conversely: if
x" is in y, then x' is not in y, so x is in y. Graph-theoretically, we represent this
as follows (see next figure):
If this is right, then it follows that an entity and the absence of its absence
‘occur’ in exactly the same set of loci: for all y, there is an inherence edge (x,
y) just in case there is an absential location edge (x", y). Can we appeal now to
a uniqueness condition for absences, and infer that the absence of an absence of
Indian Logic
381
an entity is identical to the entity? The point is controversial, with the majority
favouring identification. It is Raghunatha 71 who argues that the identification is
unsound, on the ground that nothing can turn an absence into a presence. Here
again Raghunatha’s intuition agrees with the graph-theoretic reconstruction: the
nodes x and x" are connected to other nodes by means of different types of edge.
So they cannot both represent entities of the same type. Moreover, as we shall
see in more detail below, the Naiyayikas do not accept that it is generally valid to
infer from the occurrence of x in y to the occurrence of x" there, although they
do allow the converse. This is the caveat in the line of reasoning with which I
began this paragraph. The implication is that x and x" need not, after all, share
the same set of loci.
Let us repeat the procedure once more. If x', the absence of x, is in y, then
x" is not in y, and so x'", the absence of x", is in y. Conversely: if x'" is in y,
then x" is not in y, so x' is in y. Graph-theoretically:
Ox'"
It follows that a first-order absence and the absence of its absence reside in
exactly the same set of loci. But here we can appeal to the uniqueness condition,
71 Raghunatha, Padarthatattvanirupana, p. 55. Daniel Ingalls, Materials for the Study of
Navya-Nydya Logic (Cambridge, Mass.: Harvard University Press, 1951), p. 68. Bimal. K.
Matilal, Logic, Language and Reality (Delhi: Motilal Banarsidass, 1985), p. 149. Roy W.
Perrett, “Is Whatever Exists Knowable and Nameable?” Philosophy East & West 49.4 (1999),
pp. 410-414, esp. 408-9. I disagree here with the idea of Matilal and Perrett that there is
only an intensional difference between an object and the absence of its absence. For me, a type
difference in the graph means a type difference in categories of thing.
382
Jonardon Ganeri
because the edges are all of the same type. So x"' is identical to x' , as Raghunatha
himself allows. 72 Similarly, x"" is identical to x" , and so on. There are no absential
nodes of order higher than two. The argument is summed up by Annambhatta in
the Tarkasamgraha [§89]:
The view of the early thinkers is that the absence of an absence is noth¬
ing but a presence; it is not admitted as a new absence for there would
then be an infinite regress. According to the new school, however, the
absence of an absence is a distinct absence, and there is no regress as
the third absence is identical to the first.
Recall that we defined the absence x' as a node such that x' is absentially
located in y if there is no edge between x and y. That definition was adequate for
the introduction of first-order absences, because there is only one kind of edge in
the original graph, namely the inherence edge. The expanded graph has another
sort of edge, however: the absential edge. So the notion of a second-order absence
is underdetermined by our original definition. The new definition we need is:
Rule for Absence:
An absence x' is absentially located in y if x does not inhere in y.
Rule for Higher Order Absence:
For i > 1, an absence x z is absentially located in y iff x 1-1 is not
absentially located in y.
The second clause implies that absence is a classical negation for i > 1, and so,
in particular, that an absence of an absence of an absence is identical to an absence.
A double negation, however, is a mixture — a negation defined on inherence edges
followed by a negation defined on absential qualifier edges - and for that reason
behaves non-classically. What I will show in the next section is that Navya-Nyaya
logic rejects the classical rule of Double Negation Introduction — the rule that
licenses one to infer from p to -i->p. What replaces it is a weakened rule — infer
from -ip to -i-i-i p. This is because negation is a procedure for filling ‘gaps’ in
the graph: whenever there is no edge between two nodes, the rule for negation
licenses us to insert an absential node between them. The classical rule for Double
Negation Elimination - the rule that licenses one to infer from -i->p to p — remains
valid in Nayva-Nyaya logic (i.e. if x' is not in y, then x is in y ). 73 The effect of
this weakening in the rule for Double Negation Introduction is that one is no longer
72 Paddrthatattvanirupana, pp. 67-69. Daniel Ingalls, Materials, pp. 68-69; Bimal. K. Matilal,
Logic, Language and Reality , pp. 149-150.
73 Daniel Ingalls draws a comparison between Navya-Nyaya and intuitionist logic ( Materials , p.
68, n. 135), claiming that it is the elimination rule for double negation that is rejected. However
we are able, in Navya-Nyaya logic, to infer from the absence of the absence of an entity to the
presence of that entity; conversely, we are not able to infer from the presence of an entity to the
absence of its absence — the non-pervasive node is a counter-example.
Indian Logic
383
entitled to infer that if x is in y, then x' is not in y. One effect of this is to block
the equivalence of a positive entity with the absence of its absence. We can say
that x' is the absence of x", but we cannot say that x is the absence of x'. Graph-
theoretically, connections of the form O x => O x ’ are prohibited, since a positive
entity cannot be the absence of anything. Also prohibited are triangles of the form
below, because negation behaves classically within the domain of absences. What
is stranger, however, is the effect the weakened rule has of permitting a positive
entity to be co-located with its absence. For we are no longer in a position to
assert that the presence of an entity is inconsistent with its absence. Let us see
how the Nyaya philosophers arrive at the conclusion that one must allow for such
an unusual possibility.
Qx' <S= O x "
V
Oy
Whenever something inheres in a compound substance, the question arises: does
it also inhere in the parts? An entity is said to be of ‘locus-pervading’ occurrence
just in case it inheres in all the parts of its locus (as well as in the locus itself). 74 It
saturates its locus. A sapphire is red through-and-through, and sesame oil pervades
every part of the seed; but a painted vase is blue only on the outside. Let us say
then that x is locus-pervading with respect to y just in case x inheres in y and if z
is a part of y then x inheres in z. 75 The only things that have parts are substances,
and substances inhere in their parts and in nothing else. So x is locus-pervading
with respect to y just in case x inheres in y and if y inheres in z then x inheres in
z. Certain types of quality pervade their loci, according to the classical Vaisesika
authors. 76 Examples include weight, viscosity, and fluidity. A thing is heavy just
in case every part of it is heavy. Colours, tastes, smells can pervade their loci but
need not do so. 77 And a compound substance is locus-pervading with respect to
each of its parts, if ‘part of’ is a transitive relation.
74 Ingalls (1951: 73-74); Bimal Matilal, The Navya-Nyaya Doctrine of Negation (Cambridge,
Mass.: Harvard University Press, 1968), p. 53, 72, 85; Matilal (1985: 119-122).
75 Frege’s notion of ‘divisibility’ is formally rather analogous. Gottlob Frege, The Foundations
of Arithmetic, translated by J.L. Austin (Oxford: Basil Blackwell, 1950), p. 66: “The syllables
“letters in the word three” pick out the word as a whole, and as indivisible in the sense that no
part of it falls any longer under the same concept. Not all concepts possess this quality. We can,
for example, divide up something falling under the concept ‘red’ into parts in a variety of ways,
without the parts thereby ceasing to fall under the same concept ‘red.’”
76 Karl Potter ed., “Indian Metaphysics and Epistemology - The Tradition of Nyaya-Vaisesika
up to Gangesa”, in The Encyclopedia of Indian Philosophies, vol. 2 (Delhi: Motilal Banarsidass,
1977), pp. 114-119.
77 Raghunatha, Padarthatattvanirupana, pp. 44-6.
384
Jonardon Ganeri
The notion of a locus-pervading entity has a distinctive graph-theoretic corre¬
late. An edge (ni, 112) is locus-pervading just in case there is an edge from ni to
any node in any path from 112 .
/ 1 \
O <— O n 2 —>0
While locus-pervading nodes are straightforwardly definable in the system as
so far developed, the concept of ‘unpervaded occurrence’ ( avyapya-vrttitva ) marks
a theoretical innovation. The classic Buddhist refutation of realism about wholes
is that wholes must be the bearers of contradictory properties. For if some parts
of a vase are red and other parts are not red, and if the vase as a whole has a
colour in virtue of its parts having colour, then one seems forced to admit either
that the whole is both red and not red, or that it has no colour at all. 78 The
traditional Nyaya-Vaisesika solution is less than satisfactory - it is to say that
the whole has a new shade of colour called ‘variegated’! Recognising the ad hoc
nature of such a response, later Naiyayikas try instead to make sense of the idea
that a property can be co-located with its absence. 79 The idea is to capture the
sense in which one says that the vase is red, because its surface is red, allowing
at the same time that it is not red, because its insides are some other colour. A
favourite Nyaya example involves the relation of contact: the tree enjoys both
monkey-contact (there is a monkey on one of its branches) and also the absence
of monkey-contact (its roots and other branches are in contact with no monkey).
This defence of realism is what motivates later writers to allow there to be such
a thing as unpervaded occurrence, defined to be an occurrence that is co-located
with its absence. That is, an unpervading node is a node x such that there is an
edge (x, y) and an edge (x', y). Triangles such as the following are now deemed
to be permissible in the graph:
The strangeness of such a possibility is ameliorated if one says, as some Naiyayikas
do, that x occurs in y as ‘delimited’ by one part, and its absence occurs in y as
‘delimited’ by another part. 80 Gangesa nevertheless goes to considerable lengths
to reformulate logic and the theory of inference in Navya-Nyayain a way that per-
78 DharmakTrti, Pramanavarttika II, 85-86; KamalasTla, Pahjikd under Tattvasamgraha 592-
598.
79 Udayana, Atmatattvaviveka, pp. 586-617. Prabal Kumar Sen, “The Nyaya-Vaisesika Theory
of Variegated Colour ( citrarupa ): Some Vexed Problems”, Studies in Humanities and Social
Sciences 3.2 (1996), pp. 151-172.
80 Ingalls (1951: 73-4); Bimal Matilal, The Navya-Nyaya Doctrine of Negation (Cambridge,
Mass.: Harvard University Press, 1968), pp. 71-73.
Indian Logic
385
Ox
O
/
Oy
mits the co-location of an entity with its absence. The phenomenon of unpervaded
occurrence is not regarded as a minor curiosity in Nyaya, but as the occasion for
serious revision in their analysis. 81
4-3 Definitions of logical consequence
With the introduction of absence, the graph-theoretic ontologies can serve as se¬
mantic models for a propositional language. A sentence ‘p’ is assigned, let us
stipulate, an ordered pair of nodes (x, y). The sentence is true if that pair is an
edge in the graph, false if it is not. 82 The negation of that sentence, >p’, is true
if (x', y) is an edge, false if it is not. Again, ‘-i->p’ is true if (x", y) is an edge,
false if it is not. If triangles like the one above are possible, then the truth of ‘p’
does not imply the truth of since (x, y) is an edge but not (x", y). So the
propositional logic being modelled is, as we have already observed, one in which
Double Negation Introduction does not hold. In this theory, we still have these
correspondences between truth-value and negation:
(Rl) if -■Tq then T-'Q. from Rule for Absence
(R2) T-i->q iff -iT-iq from Rule for Higher Order Absence
What we no longer have is:
(R3) if T-iq then -iTa
The reason, as I said before, is that negation is an operation that fills ‘gaps’ in
the graph - it tells us nothing when there is already an edge between two nodes.
So the truth of a proposition is consistent, in Navya-Nyaya logic, with the truth
of its negation. This element of dialetheism in the theory does not, however,
mean that anything is provable or that anything follows from anything else — the
correspondences Rl- R2 are enough to prevent the system collapsing. Let us see
why.
81 Matilal’s property-location language, in which properties have both a ‘presence range’ and
an ‘absence range’ and the two ranges are permitted to overlap, is a different way to capture the
same idea; Matilal (1985: 112-127).
82 C4angesa, Tattvacintamani , I, pramalaksana, p. 401.
386
Jonardon Ganeri
In the modern analysis of valid inference, an inference is valid just in case
it is impossible for the premises to be true without the conclusion also being
true. In the logic of classical India, validity is a matter of property-substitution,
and the problem is to determine the conditions under which the occurrence of a
reason property at a location warrants the inference that a target property occurs
there too (“ Ta because iia”). The leading idea is that such property substitutions
are valid just in case the reason does not ‘wander’ or ‘deviate’ from the target
(avyabhicara ). In a famous passage called the vyaptipancaka , Gangesa suggests
five ways to make sense of this idea: 83
Now, in that knowledge of a pervasion which is the cause of an infer¬
ence, what is pervasion? It is not simply non-wandering. For that is
not
1. nonoccurrence in loci of the absence of the target, nor
2. nonoccurrence in loci of the absence of the target which are dif¬
ferent from loci of the target, nor
3. non-colocation with difference from a locus of the target, nor
4. being the absentee of an absence which resides in all loci of absence
of the target, nor
5. nonoccurrence in what is other than a locus of the target,
since it is none of these where the target is maximal.
A ‘maximal’ property is a property resident in everything (kevalanvayin). Gangesa
dismisses the five provisional analyses on the grounds that all are formulated in
terms of ‘absence of the target’, and that that phrase is undefined when the tar¬
get is maximal (the absence of a maximal property — assumed here not to be of
unpervaded occurrence — would occur in nothing and so be ‘unexampled’, contra¬
dicting a basic condition of connectedness). In his preferred definition, Gangesa
exploits a trick to overcome this problem. 84 He says that any property whose
absence is colocated with the reason is not identical to the target. This implies
that the target is not a property whose absence is colocated with the reason, but
the contraposed formulation avoids the use of the troublesome phrase ‘absence of
the target’.
Consider now the difference between the first and second analyses in the list
of five. Graph-theoretically, what the first analysis states is that, if r is the node
representing the reason, and t is the node representing the target, then r is present
in no node where t is absent —
But what happens if the target has nonpervaded occurrence? Then the first
analysis is too strong. 85 For it is not a necessary condition on valid inference that
83 Gangesa, Tattvacintamani , II, vyaptipancaka , pp. 27-31.
84 Gangesa, Tattvacintamani, II, siddhanta-laksana, p. 100.
85 I follow here the explanation of Raghunatha. VyaptipancakadTdhiti text 3-4 (Ingalls (1951:
154)).
Indian Logic
387
Or O t' => O t
/
O
the reason not be present wherever the target is absent, if there are nodes where
the target is present as well as absent. What validity precludes is the presence
of the reason without the presence of the target. So the proper definition is that
the reason is not present wherever the target is not present (and so also absent).
This is exactly what the second analysis states. We can make the point in terms
of our earlier definitions of truth and negation. The premise in an inference is
the statement that the reason occurs in a certain location, the conclusion the
statement that the target occurs in that location. What our first analysis asserts
is that the premise is not true if the negation of the conclusion is true ( = absence
of target in the location). The second analysis states instead that the premise is
not true if the conclusion is false ( = denial of presence of target in the location).
Ironically, then, it is the very element of dialetheism of the Navya-Nyaya system
which forces Gahgesa to disambiguate the definition of validity, and to distinguish
the correct definition from the one that had been preferred before.
Let q = “the reason r inheres in x”, 0 = “the target t inheres in x”. Then
a k 0 iff t pervades r. The problem is to solve for ‘pervades’. The first solution in
Gahgesa is:
1. whatever the value of x, a is not true if ->0 is true, i.e.
a 1= 0 iff under any assignment of value to x, T~<0 -» T->a
His second solution is:
2. whatever the value of x, a is not true if 0 is not true, i.e.
a 1= 0 iff under any assignment of value to x, ->T0 —> ST a.
What we have seen is that (2) and not (1) is the correct analysis of logical
consequence if R3 is rejected. 86
4-4 Number
The classical Vaisesika theory of number is that numbers are qualities of sub¬
stances. 87 A quality ‘two’ inheres in both members of a pair of substances, another
86 R3 is what Graham Priest calls the ‘exclusion principle.’ For a semantic theory without this
principle, see his In Contradiction: A Study of the Trans consistent (Dordrecht: Martin Nijhoff
Publishers, 1987), chapter 5.
87 Vaisesikasutra 1.1.9, 7.2.1-8.
388
Jonardon Ganeri
quality ‘two’ inheres in another such pair, and all the qualities ‘two’ have inhering
in them a single universal ‘twohood’ (see graph on page 388. 88
o o
\/
0
1
/ \
o o
0 the universal “twohood”
© the quality-particular “two”
Bhasarvajna and Raghunatha, as usual, lead the reforming move. Bhasarvajna’s
theory 89 is that numbers are not qualities at all, but relations of identity and
difference. Thus the sentence ‘a and b are one’ means simply that a — 6, while
‘a and b are two’ means that a ^ b. Bhasarvajna’s analysis is echoed, very much
later, in Gadadhara’s (c. AD 1650) comments on the meaning of the word ‘one’. 90
Gadadhara states that the meaning of ‘one F’ is: an F as qualified by being-
alone, where ‘being alone’ means ‘not being the absentee of a difference resident in
something of the same kind.’ In other words, ‘one F’ is to be analysed as saying
of something which is F that no F is different to it. If this is paraphrased in a
88 For a more detailed description of the classical account: Jonardon Ganeri, “Objectivity and
Proof in a Classical Indian Theory of Number,” in Synthese, 129 (2001), pp. 413-437.
89 Nyayabhusana, p. 159.
90 Saktivada with Krsna Bhatta’s Manjusa, Madhava Bhattdcarya’s Vivrtti and Sahitya
Darsandcarya’s Vinodini, edited by G. D. Sastri (Benares: Kashi Sanskrit Series no. 57, 1927).
p. 189.
Indian Logic
389
first-order language as l Fx & -i(3 y){Fy k y ^ x)’ then it is formally equivalent
to a Russellian uniqueness clause ‘Fx k ( \fy)(Fy —> y = x)\ The idea that
‘one’ expresses uniqueness is in the spirit of Bhasarvajna’s idea that it denotes the
identity of a thing. In any case, it is clear that, for Gadadhara, ‘one’ has a logical
role similar to that of the definite article.
Raghunatha is more radical still. 91 The central problem is that things in any
category in the Vaisesika ontology can be numbered, and Raghunatha concludes
that numbers must belong in a new category of their own:
Number is a separate category, not a kind of quality, for we do judge
that there is possession of that [number] in qualities and so on. And this
[judgement we make that qualities have number is] not an erroneous
one, for there is no [other] judgement which contradicts it.
Raghunatha puts pressure at exactly the right place. The ‘is-the-number-of’
relation is not reducible to the relation of inherence or any relation constructed
out of it, for it is a relation between numbers and any type of thing. What is
this new relation? Raghunatha points out that while inherence is a distributive
relation (avyasajya-vrtti) , the number-thing relation has to be collective ( vyasajya-
vrtti). The distinction occurs in the context of sentences with plural subjects. An
attributive relation is distributive if it relates the attribute to every subject —
if the trees are old, then each individual tree is old. A relation is collective if it
relates the attribute to the subjects collectively but not individually — ‘the trees
form a forest’ does not imply that each tree forms a forest. Number attributions
are collective; if one says that there are two pots here, one does not imply that
each pot is two. Inherence, however, is a distributive relation, and so cannot be
the relation of attribution for numbers. This new relation is called the ‘collecting’
(paryapti ) relation by Raghunatha: 92
The collecting relation, whose existence is indicated by constructions
such as “This is one pot” and “These are two”, is a special kind of
self-linking relation.
His commentator JagadTsa explains:
It might be thought that the collecting relation is [in fact] nothing but
inherence...So Raghunatha states that collecting [is a special kind of
self-linking relation]. ... In a sentence like “This is one pot”, collecting
relates the property pot-hood by delimiting it as a property which
resides in only one pot, but in a sentence like “These are two pots”,
collecting relates the property twoness by delimiting it as a property
which resides in both pots. Otherwise, it would follow that there is
no difference between saying “These are two” and “Each one possesses
twoness”.
91 Padarthatattvanirupana, pp. 86-87.
92 Avacchedakatvanirukti with JagadTsa’s Jagadtsi, edited by Dharmananda Mahabhaga
(Varanasi: Kashi Sanskrit Series 203), p. 38.
390
Jonardon Ganeri
Thus the number two is related by the collecting relation to the two pots jointly,
but not to either individually. Raghunatha’s idea is clear in the graph-theoretic
context. The introduction of numbers requires one final expansion of the graph.
We introduce another new domain of nodes (1, 2, 3,.. ) and another new type of
edge from these nodes. Like ordinary edges, this new type of edge is an ordered
pair whose first member is a node, but now the second member is set of nodes.
The new edge connects the node 2 with every pair of nodes (x, y). Likewise, it
connects the node 3 with every triple of nodes (x, y, z), and so on. The node
2, then, is that node from which all edges to pairs begin, the node 3 the node
from which all edges to triples begin, and so forth. This is enough to individuate
number-nodes graph-theoretically (see graph on 390:
The nodes to which the new edge can connect a number-node can be of any
type. In particular, they can themselves be number-nodes. Indeed, the new edge
connects 2 with pairs of nodes one of whose members is 2 itself (see graph on page
391:
This solves the cross-categorial problem. Number-nodes are related by the new
kind of bifurcating edges to nodes of any and every type in the graph, including
number-nodes themselves.
The graph-theoretic approach is, I think, full of potential. It offers a new way to
read and interpret Navya-Nyaya logic. One might proceed by looking for further
situational constraints on what constitutes a permissible graph and applying graph
theory to analyse the structure of those graphs. One might also try to establish
Indian Logic
391
O
the relationship between such graphs and classical or nonclassical logics. The
treatment of negation suggests a comparison with dialetheic logic, and the idea
of self-linking nodes perhaps with non-wellfounded set theory. My aim here has
been to expose the logical basis of Vaisesika theory, and to draw a conclusion
about the nature of logical thinking in India. The conclusion is simply this. The
idea that nature instantiates mathematical structure is not remote from the Indian
understanding of natural philosophy, contrary to what has generally been believed,
but is in fact a fundamental aspect of it.
BIBLIOGRAPHY
Selected Indian Logical Texts
[Mogalliputta Tissa, c. 3rd BC] Mogalliputta Tissa (c. 3 rd BC). Kathavatthu. Translation -
Aung (1915). Discussion - Bochenski (1956), Ganeri (2001), Matilal (1998: 33-7), Schayer
(1932-33).
[Milinda-panha, c. 1st AD] Milinda-panha (c. 1st AD). Translation - T. W. Rhys Davids (1890).
[Agnivesa, c. 100 AD] Agnivesa (c. 100 AD). Carakasamhita. Translation - Sharma (1981-94).
Discussion - Gokhale (1992), Matilal (1998: 38-43), Prets (2000), Solomon (1976, chapter 2).
[Kanada, c. 200 AD] Kanada (c. 100 AD). Vaisesikasutra. Translation - Sinha (1911). Discus¬
sion - Nenninger (1994), Nozawa (1991), Schuster (1972)
[Nagarjuna, c. 200 AD] Nagarjuna (c. 200 AD). Vaidalyaprakarana, Translation - Tola & Drag-
onetti (1995).
[Nagarjuna, c. 200 AD] Nagarjuna (c. 200 AD). Upayahrdaya. Discussion - Tucci (1929b).
[Gautama, c. 150 AD] Gautama Aksapada (c. 150 AD - 250 AD). Nydyasutra. Translation -
Gangopadhyay (1982). Discussion - Bochenski (1956), Chakrabarti (1977), Ganeri (2000),
Ganeri (2001), Gokhale (1992), Matilal (1985), Matilal (1998), Prets (2001), Schayer (1933),
Randle (1924),
392
Jonardon Ganeri
[Vasubandhu, c. 400 AD] Vasubandhu c. 400 AD - 480 AD). Vadavidhi, Vadavidhana,
Tarkasastra. Discussion - Tucci (1929a), Tucci (1929b).
[Vatsyayana, c. 350 AD] Vatsyayana (c. 350 AD - 425 AD). Nyayabhasya. Translation - Gan-
gopadhyay (1982). Discussion - Bochenski (1956), Matilal (1998).
[Dinnaga, c. 480 AD] Dinnaga (c. 480 AD - 540 AD). Pramanasamuccaya. Translation - Hayes
(1988). Discussion - Bochenski (1956), Ganeri (2001), Hayes (1980), Hayes (1988), Herzberger
(1982), Katsura (1983), Katsura (1986a), Matilal (1998), Matilal k Evans eds. (1986), Oetke
(1994).
[Dinnaga, c. 480 AD] Dinnaga (c. 480 AD - 540 AD). Hetucakranirnaya. Translation - Chatterji
(1933), Chi (1969). Discussion - Bharadwaja (1990), Bochenski (1956), Chi (1969), Randle
(1924)
[Sankarasvamin, c. 500 AD] Sankarasvamin (c. 500 AD - 560 AD). Nyayapravesa. Translation
- Tachikawa (1971). Discussion - Chi (1969), Gillon k Love (1980), Oetke (1996).
[Uddyotakara, c. 500 AD] Uddyotakara (c. 550 AD - 625 AD). Nyayavarttika. Translation -
Jha (1984). Discussion - Gokhale (1992).
[DharmakTrti, c. 600 AD] DharmakTrti (c. 600 AD - 660 AD). Pramdnavarttika. Discussion -
Gokhale (1992), Hayes (1987), Katsura ed. (1999), Matilal (1998), Matilal k Evans eds.
(1986), Steinkellner (1973), Steinkellner ed. (1991).
[DharmakTrti, c. 600 AD] DharmakTrti (c. 600 AD - 660 AD). Nyayabindu. Translation - Gan-
gopadhyay (1971), Stcherbatsky (1930, volume 2). Discussion - Gokhale (1992).
[DharmakTrti, c. 600 AD] DharmakTrti (c. 600 AD - 660 AD). Vadanyaya. Translation -
Gokhale (1993). Discussion - Chinchore (1988).
[Siddhasena, c. 700 AD] Siddhasena (c. 700 AD). Nyaydvatara. Translation and discussion -
Balcerowicz (2001).
[Udayana, c. 1050 AD] Udayana (c. 1050 AD). Nydyavdrttikatdtparyaparisuddhi,
Nydyaparisista, Laksanavali.
[Gangesa, c. 1325 AD] Gangesa (c. 1325 AD). Tattvacintamani. Discussion - Bhattacharyya
(1987), Bochenski (1956), Gangopadhyay (1975), Goekoop (1967), Ingalls (1951), Matilal
(1968), Matilal (1985), Matilal (1998), Staal (1988), Vattanky (2001), Wada (1990), Wada
(forthcoming).
Secondary Literature on Indian Logic
[Aung, 1915] S. Z. Aung. Points of Controversy, or, Subjects of Discourse: Being a translation
of the Kathavatthu from the Abhidhammapttaka, eds. S.Z. Aung and C.A.F. Rhys Davids.
Pali Text Society. Routledge and Kegan Paul, London, 1915.
[Bagchi, 1953] S. Bagchi. Inductive Reasoning: A Study of Tarka and its Role in Indian Logic.
Munishchandra Sinha, Calcutta, 1953.
[Balcerowicz, 2001] P. Balcerowicz. Epistemology in Historical and Comparative Perspective:
Critical Edition and English Translation of Logical-Epistemological Treatises: Nyaydvatara,
Nydyavatara-vrti and Nydyavatdra-t ippana with Introduction and Notes, Franz Steiner Ver-
lag, Hamburg, 2001.
[Balcerowicz and Mejor, 2000] P. Balcerowicz and M. Mejor, eds. On the Understanding of other
cultures: Proceedings of the International Conference on Sanskrit and Related Studies to
Commemorate the Centenary of the Birth of Stanislaw Schayer, Warsaw 1999. Oriental In¬
stitute, Warsaw University. 2000.
[Bharadwaja, 1990] V. Bharadwaja. Form and Validity in Indian Logic. Indian Institute of
Advanced Study, Shimla, 1990.
[Bhattacharyya, 1987] S. Bhattacharyya. Some Aspects of the Navya-Nyaya Theory of Infer¬
ence, In Doubt, Belief and Knowledge, S. Bhattacharyya, pp. 245-267. Indian Council of
Philosophical Research, Delhi, 1987.
[Bochenski, 1956] J. M. Bochenski. The Indian Variety of Logic. In A History of Formal Logic,
2nd edn, J. M. Bochenski, pp. 416-447. Trans. I. Thomas, Chelsea Publ. Co., New York, 1961.
[Chakrabarti, 1977] K. K. Chakrabarti. The Logic of Gotama. University of Hawaii Society for
Asian and Comparative Philosophy Monograph, no. 5. University Press of Hawaii, 1977.
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[Chatterji, 1933] D. Chatterji. Hetucakranirnaya - A Translation. Indian Historical Quarterly,
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This Page Intentionally Left Blank
THE MEGARIANS AND THE STOICS
Robert R. O’Toole and Raymond E. Jennings
1 INTRODUCTION
In the opinion of Carl Prantl, the nineteenth century historian of logic, the Stoics
were no logicians at all, but merely confused plagiarists who peddled second-rate
versions of Peripatetic and Megarian doctrines. Chrysippus of Soli, touted by
many as the greatest logician of the Hellenic age, was a special target of Prantl’s
baleful attacks, as witness the following assessment of his logical skills:
Chrysippus created nothing really new in logic, for he only repeated
details already known to the Peripatetics or pointed out by the Megar-
ians; his activity consisted in this, that in the treatment of the material
he descended to a pitiful degree of dullness, triviality, and scholastic
quibbling. ... It is to be considered a real stroke of luck that the works
of Chrysippus were no longer extant in the Middle Ages, for in that
extensive morass of formalism, the tendency (weak as it was) toward
independent investigation would have been completely eliminated. 1
Although Chrysippus may have borne the main thrust of Prantl’s assault, there
is no doubt that his criticisms were meant to have general application among the
philosophers of the Stoa.
It is difficult to discern the motivation behind Prantl’s ad hominem attacks on
the Stoics in general and on Chrysippus in particular, and one has to look back
to Plutarch’s polemic in De communibus notitiis and De Stoicorum repugnantiis
to find anything remotely similar in tone. On the other hand, it is not so difficult
to see, as Mates argues [Mates, 1953, pp. 87-88], that one can safely discount
virtually all of Prantl’s judgements on Stoic logic, not only because there is lacking
any argument to substantiate them, but also because Prantl himself seems to have
had little understanding of what the Stoics were about (cf. Bocheriski 1963, 5).
Unfortunately, Prantl’s estimation of Stoic logic, echoed by Eduard Zeller with
somewhat more scholarly decorum, but with little more skill and understanding,
was to remain for the most part unchallenged until late in the first half of the
twentieth century when it was called into question, first, by Jan Lukasiewicz, and
later, by Benson Mates.
1 Geschichte der Logik im Abendlande, 1.404. Translated by Mates [1953, p. 87].
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
© 2004 Elsevier BV. All rights reserved.
398
Robert R. O’Toole and Raymond E. Jennings
Prantl’s work, according to Lukasiewicz, although “indispensable ... as a collec¬
tion of sources and material,... has scarcely any value as an historical presentation
of logical problems and theories” [Lukasiewicz, 1967, p. 67]. Moreover, since nei¬
ther Prantl, nor Zeller, nor any other of the older historians of logic, had any
understanding of the difference between the logic of terms, which was Aristotle’s
logic, and the logic of propositions, which was the logic of the Stoa, there exists no
history of the logic of propositions, and thus, no complete understanding of the his¬
tory of formal logic. For this reason, “the history of logic must be written anew,
and by an historian who has fully mastered mathematical logic” [Lukasiewicz,
1967, p. 67].
In the interest of setting the historical record straight, one cannot help but
endorse this project, and we ought to be ever grateful not only to Lukasiewicz,
but also to Mates, Bochenski, the Kneales, and others for their contributions
toward this end. It should be noted, however, that there is a danger that the
historian of logic possessing this requisite mastery of mathematical logic may allow
his or her familiarity with the discipline to obscure, or even distort, the historical
enterprise. When viewing the past from the perspective of contemporary doctrines,
it is sometimes all too easy to succumb to the appeal of a ‘convergence’ theory of
history, and to assume that one’s predecessors, if only they had got it right, would
have come to the same place we now occupy. At any rate, there seems to have
been a tendency toward such a view among several modern commentators. These
writers seem to presume that the Stoics had in mind to develop a formal logic along
the lines of the modern propositional calculus, and their respective appraisals of
Stoic logic might be seen as dependent on their estimates of the degree of success
to which this goal was carried through.
It must be admitted, however, that there are some texts which would seem to
justify such a presumption; on the other hand, all of these texts can be, and have
been, called into question. To consider two examples, there are texts which may
be taken to indicate that, in general, the Stoics defined those logical constants
having a role in their syllogistic as binary connectives; other texts may be taken
to support the view that Stoic logic was a formal logic—formal, that is, in the
specialised sense in which mathematical logic is formal, namely, the substitutional
sense.
As to the first of these suppositions, it is surely false that the logical connectives
which appear in the Stoic syllogisms were in general defined as binary connectives.
In particular, consider SieCsuypevov , which is evidently the notion of disjunction
occurring in the fourth and fifth Stoic syllogisms. 2 This disjunction, represented
at the linguistic level by the connective particle ‘fj’ (or ‘rjToi’), is assumed by
not a few writers to be the ‘exclusive’ disjunction defined by the matrix 0110. 3
‘The fourth syllogism may be represented by the schema l a or /3; but a; therefore, not /3’,
and the fifth by the schema ‘a or /3; but not a; therefore, /3’.
3 Lukasiewicz [1967, p. 74]; Bochenski [1963, p. 91]; Mates [1953, p. 51]; Kneale and Kneale
[1962a, p. 148]; Gould [1970, p. 72],
The Megarians and the Stoics
399
The reason for this assumption, according to some writers at least, 4 is that an
inclusive notion of disjunction will not support both the fourth and fifth Stoic
syllogisms, whereas the exclusive disjunction will. It can be shown, however, that
SieCEuypevov, the Greek notion of disjunction which validates both the fourth and
the fifth syllogisms, is not the exclusive disjunction of modern logic in which the
connective is defined as a binary operator satisfying the matrix 0110.
Now it is not at all clear whether the Stoics viewed SieCeuypevov as a purely
truth-functional notion of disjunction, but it seems evident from the texts which
mention a disjunction consisting of more than two disjuncts that if the Stoic dis¬
junction were to be characterized truth-functionally, then the general truth con¬
dition (i.e. the truth condition for the occurrence of two or more disjuncts) would
be that it is true whenever exactly one of its clauses is true. 5 On the other hand, it
can be shown by means of a simple inductive proof that the general truth condition
for the modern exclusive disjunction is that an odd number of disjuncts be true.
As a consequence of this result, it is apparent that even if the Stoic disjunction
is given a truth-functional interpretation, its truth will coincide with that of the
0110 disjunction only in the two-disjunct case.
It seems evident, then, that we can rule out the assumption that the Stoic
disjunction and the modern exclusive disjunction are identical; hence, we can not
take their alleged identity as evidence for the conclusion that the Stoic disjunction
was viewed as a binary connective. Furthermore, there seems to be nothing in the
texts to force the interpretation of StsCeuypevov as even having a fixed ‘arity’, 6
and in particular, nothing to force the interpretation that it has arity two. But
if a Stoic disjunction has no fixed arity, then we are not required—indeed, we are
not allowed —to treat an n-term disjunction as a two-term disjunction by the use
of some bracketing device. This proscription has consequences for the idea that
Stoic logic was formal in the substitutional sense, an issue which we shall take up
later on in this article.
If it is neither true that the logical connectives occurring in the Stoic syllogisms
are in general binary operators, nor that Stoic logic is a formal system in the
substitutional sense of formal, then the modern historians and commentators who
have affirmed the contrary have misunderstood and misrepresented Stoic logic
no less than have the earlier historians of logic, such as Prantl and Zeller. The
obstacle for both the later and earlier writers, it seems to us, is that they have
allowed their preconceptions to obscure their understanding. It is obvious from
their writings that Prantl and Zeller held to the general conviction of their era that
the intellectual achievements of the Hellenic period in Greece were few indeed.
And no doubt this view would have affected their ability to provide a balanced
account of Stoic philosophy in general, and Stoic logic in particular. As for the
writers who published in the early and middle years of this century, they no doubt
4 Mates [1953, p. 52); Bocheriski [1963, p. 9l).
5 cf. Aulus Gellius Nodes Atticae. 16.8.13-14; Galen inst. log. 12; PH 2.191.
6 The ‘arity’ of a binary operator is 2; that of a ternary operator, 3; of a quaternary operator,
4; and so on.
400
Robert R. O’Toole and Raymond E. Jennings
were influenced by the tendency, experienced at some time or other by most of us
familiar with modern formal logic, to suppose that there really is no other notion
of logic.
Be that as it may, we are nevertheless indebted to these logicians and historians
of logic for having rescued the logic of the Stoa from the lowly status to which it
was relegated at the hands of Prantl and his contemporaries. That having been
said, it needs also to be said that we ought now to move out from the shadows of
Lukasiewicz, Bochehski, and Mates, and attempt an interpretation of Stoic logic
less coloured by a reverence for modern formal systems, and more in harmony with
what the texts seem to indicate as being the place of logic in the Stoic system as
a whole. This point of view is well expressed by Charles Kahn:
We may not have an accurate picture of Chrysippus’ enterprise in “di¬
alectic” if we see it simply as a brilliant anticipation of the propositional
calculus. No doubt it could not be accurately seen at all until it was
seen in this way, again by Lukasiewicz and then more fully by Mates.
But now that their insights have been assimilated, I think it is time
to return to a more adequate view of Stoic logic within the context of
their theory of language, their epistemology, their ethical psychology,
and the general theory of nature [Kahn, 1969, p. 159].
The elements of Stoic philosophy mentioned by Kahn— epistemology, theory of
language, ethical psychology, and a general theory of nature—are just the elements
viewed as extraneous to the logician’s enterprise by the early modern commenta¬
tors. Indeed, the creation of contemporary formal logic by Frege required, in the
words of Claude Imbert, “[the] gradual and piecemeal disintegration of a logical
structure built by or borrowed from the Stoics” [Imbert, 1980, p. 187]. On this
account, it seems evident that any attempt to understand Stoic logic as a formal
calculus must fail; moreover, it would seem that anyone wishing to provide an
adequate understanding would be constrained to do so as part of a reconstruc¬
tion of the logical edifice built by the Stoics. The present essay is one attempt to
formulate such an interpretation.
Leaving aside the matter of setting straight the historical record, one might ask
what the worth of studying an ancient logic such as that of the Stoics might be.
The answer to this question, it seems to us, lies in how one views the nature of
Stoic logic itself. For our part, we believe that Stoic logic developed out of a desire
to provide an account of the inferences one could make concerning the natural
course of events, such inferences depending on premisses based in the perceptual
knowledge of the occurrence of particular events or states of affairs, and in a
general knowledge of relationships discovered in nature between events or states
of affairs of certain types.
The relationships between events or states of affairs which the Stoics referred
to as ‘consequence’ (dxoXouflla) and ‘conflict’ (payr]), are represented in the Stoic
syllogistic in the major premisses of four of their five basic inference schemata. Par¬
ticular events or states of affairs are represented in the minor premisses. Knowl-
The Megarians and the Stoics
401
edge of these relationships and particular events is based on certain perceptual
structures called ‘presentations’ (cpavxaatat). Associated with these presentations
as their content are conceptual structures called pragmata (Ttpaypaxa), and asso¬
ciated with the pragmata are ‘propositional’ structures called lekta (Aexxa). Ac¬
cording to the Stoic theory, we proceed from language and thought to the world,
and to language and thought from the world, through the media of these various
structures.
This theory suggests a different paradigm from that of present-day formal logic.
It would seem to imply an understanding of logic as a human linguistic practice—a
theory of inference rather than of inferability. Given the difficulties encountered
so far in attempts to develop automated inference systems based on the paradigm
of modern formal logic, it may be worthwhile to attempt a formalisation of the
Stoic semantic theory in the hope that such a formalisation would provide a more
successful alternative. The first step in such an enterprise would be to try to
develop as clear an understanding of the Stoic theory as is possible.
2 HISTORICAL SURVEY
On the assumption that more than a few readers will be unfamiliar with early Stoic
philosophy, and since this essay is an interpretation of certain logical doctrines of
the Old Stoa, it would seem appropriate to present first a brief historical sketch
of the Stoic School, and, in particular, of the philosophers of the early Stoa.
2.1 The influence of Stoicism
The first Stoic was Zeno of Citium who founded the school some time near the turn
of the century between the third and fourth centuries B.C. The last Stoic, according
to Eduard Zeller [Zeller, 1962, p. 314], was Marcus Aurelius, the Roman emperor
who died in 180 A.D. Of course, as J. M. Rist points out [Rist, 1969c, p. 289], one
ought to understand this claim not in a literal sense, for there were Stoics who
came after Aurelius, but rather in the sense that with his passing the school came
to an end as a recognisable entity. Hence the Stoic School was extant for a period
of almost five hundred years, a remarkable achievement by any standard. But it is
not only the longevity of the school which would seem exceptional, for it might be
claimed, as at least one scholar has done, that “Stoicism was the most important
and influential development in Hellenistic philosophy” (A. A. Long [1986], 107).
The basis for this claim lies in part at least in the far-reaching domain of Stoic
doctrines. For according to Long, not only were Stoic teachings prevalent among
a large segment of the educated population in the Hellenic era, but also their
influence is apparent in various intellectual spheres during the early post-Hellenic
period as well as from the Renaissance up to fairly recent times. The tenets of
Christian philosophy, for example, exhibit certain evidence of Stoic bias, as do the
moral precepts of Western civilisation in general. Moreover, the manifestations of
such influence would seem apparent in the realm of secular literature and thought
402
Robert R. O’Toole and Raymond E. Jennings
as well, reaching a peak, according to Long, between 1500 and 1700 (1986, 247).
One scholar even perceives the presuppositions of Stoic logical principles at work
in the writings of the Alexandrian novelists and poets (Claude Imbert [1980]). In
later philosophy, according to Long, Stoic canons are evident in the writings of
men as diverse in their beliefs as the religious philosopher Bishop Butler and the
metaphysicians, Spinoza and Kant (1986, 107). And according to Imbert: “[The]
gradual and piecemeal disintegration of a logical structure built by or borrowed
from the Stoics was a necessary preliminary to Frege’s formulation of a sentential
calculus and to the conception ... of such a calculus as an independent system”
[imbert, 1980, p. 187].
But if Stoicism has indeed found expression in such widespread and various areas
of thought, then why, one might ask, does it seem that in the English-speaking
world at least, Stoic philosophical theories have been so little studied in recent
times as compared with the treatises of Plato and Aristotle—aside, that is, from
the chiefly moralistic writings of the later Stoics such as Marcus Aurelius. The
answer no doubt lies, at least in part, in the fact that virtually none of the records of
the early Stoics has survived the vagaries of time. We are fortunate to have mined
in their extant writings an abundant source from which we may develop a deep
appreciation of the philosophical thought of Plato and Aristotle, but there is no
mother lode of philosophical literature from which to develop a rich understanding
of Stoic thought. Those direct sources which have survived consist in a few badly
damaged papyri salvaged from the ruins of Herculaneum. For the most part,
however, one must rely on the reports of various commentators of uneven reliability,
authors such as Sextus Empiricus, 7 Diogenes Laertius, 8 Galen, Cicero, Stobaeus,
Plutarch, Alexander of Aphrodisias, and Aulus Gellius, to name a few. Many
of these reports are presented in the form of doxography, and many are second¬
hand; some, however, possess the merit of having direct quotations appearing in
them, and it is from these quotations that the most reliable information can be
gleaned. But even having quotations available is no guarantee that one is getting
an unbiased account. For much of the commentary by authors such as Plutarch,
7 Sextus Empiricus ( circa A.D. 200) was a Greek physician and sceptic philosopher who suc¬
ceeded Herodotus of Tarsus as head of the Sceptic School. Sextus’ critique of Stoic philosophical
doctrine is covered in a series of eleven books under the general title of Adversus Mathematicos
(abbreviated AM), and a series of three books under the title Outlines of Pyrrhonism (abbre¬
viated PH). Stoic logical doctrine is contained for the most part in Books 7 and 8 of AM and
in Book 2 of PH. Physical tenets are covered in Books 9 and 10 of AM and in Book 3 of PH.
Ethical teachings are criticised in Book 11 of AM and in Book 3 of PH. Sextus’ account of Stoic
Philosophy is probably one of the most extensive of the ancient commentaries. However, because
of it polemic nature, its value is perhaps less than it might been.
8 After Sextus Empiricus, Diogenes Laertius (circa A.D. 200-250?) provides the next most
extensive account of Stoic doctrine. Much of what he writes on the Stoics corroborates what
is written by Sextus Empiricus, but in contrast to the latter’s account, Diogenes’ report has
the advantage that it is not in the least polemical. It is fortunate that in Book 7 of Lives and
Opinions of Eminent Philosophers, the section in which he covers the Stoic School, Diogenes
draws on a handbook written by Diodes of Magnesia. His account of Stoic logic, therefore, is
widely considered to be reliable. Since he is not in general considered a reliable source, it might
have been otherwise.
The Megarians and the Stoics
403
Alexander, Sextus Empiricus, Cicero, and to some extent, Galen, is polemical in
tone and clearly inimical to various Stoic doctrines. Hence, one suspects that
quotations are often chosen not to illustrate a point in a positive mode, but rather
to show up perceived absurdities and inconsistencies in Stoic thought. Given
problems of this kind, it is clear that the road to an understanding of Stoicism
could not be as free from impediment as that to an understanding of Plato and
Aristotle.
A further reason why Stoic philosophy has been by comparison so little studied
in recent times might be the bad press it received at the hands of the nineteenth
century historians of philosophy, Prantl and Zeller. 9 Prantl apparently took to
heart Kant’s pronouncement that “since Aristotle ... logic has not been able to
advance a single step, and is thus to all appearance a closed and completed body
of doctrine.” 10 Hence, in his Geschichte der Logik , he was especially critical of
Stoic logical doctrines, attacking with a vehemence curiously out of place in what
is supposed to be a scholarly work, not only these doctrines themselves, but also
the men who authored them. Nor did Stoic logic fare much better at the hands
of Zeller, although it must be admitted that his critique lacks to some extent
the intense personal animosity characteristic of Prantl’s writings. According to
Zeller, “no very high estimate can be formed of the formal logic of the Stoics”
[Zeller, 1962, p. 123], and “the whole contribution of the Stoics in the field of
logic consists in ... clothing the logic of the Peripatetics with a new terminology”
[Zeller, 1962, p. 124], But it was not only the field of Stoic logic that received
such negative assessment from Zeller, for in his judgement the Hellenistic era was
characterised by a general decline in the quality of intellectual life, and by the
particular decline in the virtue of the philosophical enterprise (cf. Long [1986,
p. 10; 247]). There would seem to be little doubt, according to Long, that the
estimates of Prantl and Zeller in favour of Platonic and Aristotelian philosophy
and against Hellenistic philosophy carried a good deal of weight among succeeding
historians of the subject [Long, 1986, p. 10]. Perhaps no further reason than this
need be sought to account for the want of a general interest in Stoic philosophy in
the modern era.
2.2 The philosophers of the Early Stoa
Traditionally, the Stoic era has been divided into three periods: the Early Stoa
(often the Old Stoa), which begins with Zeno and ends with Antipater; the Middle
Stoa, which covers the leadership of Panaetius and Posidonius; and the Late Stoa,
which is represented by Seneca, Epictetus, and Marcus Aurelius. 11 Since this
9 We have relied on the commentaries of Benson Mates [1953, pp. 86-90] and 1. M. Bocheriski
[1963, pp. 6-8]; [1963, pp. 5-6] for an assessment of Prantl’s contribution to the history of Stoic
logic.
10 Critique of Pure Reason, Unabridged Edition. Translated by Norman Kemp Smith. Macmil¬
lan, 1929. New York: St. Martin’s Press, 1965.
n See, for example, A. A. Long, [1986, p. 115]; Jonathan Barnes, Oxford History of the
Classical World, p. 368.
404
Robert R. O’Toole and Raymond E. Jennings
essay will be concerned mainly with the logical doctrines of the Early Stoa, and in
particular the doctrines of Zeno, Kleanthes, and Chrysippus, we shall have little
to say in this introduction about the Middle and Late periods of Stoic history;
indeed, given these restrictions, we shall not have much to say about those Stoics
such as Diogenes of Babylon, Antipater of Tarsus, and several others, who came
after Chrysippus but yet belong to the Old Stoa. Furthermore, the scope of this
introduction will not permit more than a cursory glimpse of the lives of Zeno and
his two immediate successors, nor will it allow more than a brief and somewhat
arbitrary survey of their philosophical doctrines.
Zeno (333-261 B.C.)
As mentioned above, the Stoic School was founded by Zeno, a native of Citium on
the island of Cyprus. Born around 333/2 B.C., Zeno is reported to have arrived
in Athens at about the age of twenty-two and to have begun his teaching about
300/301 B.C. 12 By way of establishing himself as a teacher, he chose to discourse
with his followers in the ‘Painted Colonnade’ (Stoa Poikile ), where Polygnotus’
paintings of the Battle of Marathon were displayed. Hence, his followers, known
first as the ‘Zenonians’, came to be known as ‘the men of the Stoa’ or ‘the Stoics’
(DL 7.5).
In the period between his arriving in Athens and his starting the Stoic School,
Zeno engaged in the development of his formal philosophical education. He ap¬
parently began this training by studying with the Cynic philosopher Crates (DL
7.4), but though it would seem fairly certain that Zeno’s first formal training in
philosophy was in the doctrines of the Cynics, it is possible that he had prepared
himself for the study of philosophy by reading the many books on Socrates which
his father, a merchant, had brought home from his trips to Athens (DL 7.31). After
the period of study with Crates, Zeno became a pupil of the Megarian philosopher
Stilpo and of the Dialectician Diodorus Cronus (DL 7.25). As well, he “engaged
in careful dispute” with the Dialectician Philo of Megara, who was also a student
of Diodorus Cronus (DL 7.16). He is further reported to have studied with the
Academic philosophers Xenocrates and Polemo (DL 7.2), but given the chronology
cited above, there is some doubt that he actually did study with the former (cf.
Zeller [1962, 37nl]). On the other hand, there is confirmation that he was a pupil
of Polemo, for Cicero explicitly states that this was so (de fin. 4.3), and Diogenes
Laertius attests that he was making progress with Diodorus in dialectic when he
would enter Polemo’s school (7.25). It may be, however, that his attendance at
12 Much of the information on the lives and doctrines of the early Stoics come from Diogenes
Laertius’ Lives and Opinions of Eminent Philosophers. However, we have also relied heavily on
Reichel’s Stoics, Epicurians, and Sceptics, which is translated from the third volume of Zeller’s
Die Philosophic der Griechen, as well as on Long’s Hellenistic Philosophy, on Long and Sedley’s
The Hellenistic Philosophers, on Sandbach’s The Stoics, on Rist’s Stoic Philosophy, and on
Barnes’ article in The Oxford History of the Classical World, pp. 365-85.
It is evident from Diogenes account that there was some dispute concerning the important
dates in Zeno’s life. By way of reconciling this problem, we will report the alternatives which
Long and Zeller suggest to be the most reliable.
The Megarians and the Stoics
405
Polemo’s lectures was somewhat surreptitious, for according to Diogenes, Polemo
is said to have reproached him thus: “You slip in, Zeno, by the garden door—I am
quite aware of it—you filch my doctrines and give them a Phoenician make-up.” 13
However that may be, it is in light of Cicero’s further testimony that Zeno’s associ¬
ation with Polemo is significant, for he maintains that Zeno had adopted Polemo’s
teaching on the primary impulses of nature (de fin. 4.45), as well as on the doctrine
that the summum bonum is ‘to live in accordance with nature’ (de fin. 4.14).
There is some controversy among modern scholars whether the Peripatetics had
any influence on Zeno’s philosophical education, and, assuming they did have, in
what it might have consisted. Some commentators propose that there was such in¬
fluence, 14 but it would seem that such proposals should be regarded as conjecture,
for unlike the situation with the Cynics, the Megarians, the Dialecticians, and
those from the Academy, “there is no ancient evidence concerning Zeno’s relation¬
ship with Theophrastus or other Peripatetics” Long [1986, p. 112]. 15 In addition,
many writers insist that certain Stoic doctrines must have developed either as an
extension of Aristotelian theories, or, as a reaction to them, and underlying this
contention is the assumption that the early Stoics must have had available, and
made a close study of, the corpus of Aristotelian literature which we have available
to us. 16 In his monograph “Aristotle and the Stoics,” F.H. Sandbach conducts a
careful study of these claims and of the assumptions which underlie them. He con¬
tends that this investigation supports the theory that, barring a few exceptions,
this corpus of Aristotle’s works (which he refers to as the ‘school-works’) was not
available to the early Stoics [Sandbach, 1985, p. 55]. On the other hand, where
certain Stoic doctrines are similar in content to passages in the school-works, this
similarity may be explained by the hypothesis that an analogous passage occurred
in an ‘exoteric work’, 17 and some Stoic had read it there. Alternatively, the early
Stoics may have come across these teachings in oral form, either as explicitly at¬
tributed to Aristotle or as recognised Peripatetic doctrine. Another possibility,
one apparently acknowledged by few, if any, writers, is that the Stoics may simply
have thought of these doctrines independently of Aristotle [Sandbach, 1985, p.
55]. Sandbach concludes that for the most part the evidence will not support the
probability, let alone the certainty, that Aristotle was an influence on the origin
13 Sandbach: “... Citium, once a Greek colony, was [at the time of Zeno’s birth] predominantly
Phoenician in language, in institutions, and perhaps in population. Zeno’s contemporaries who
called him a Phoenician may have been justified in so doing, but he must be imagined as growing
up in an environment where Greek was important” [Sandbach, 1975, p. 20].
14 Andreas Graeser, for example, in his Die logischen Fragmente des Theophrast assumes that
Zeno was a pupil of Theophrastus (44).
15 Cf. F.H. Sandbach’s reference “the striking absence of explicit evidence that the early Stoics
took an interest in the work of Aristotle or of his following in the Peripatos” [Sandbach, 1985,
p. 55].
16 This view is expressed by Sandbach in his concluding paragraph [Sandbach, 1985, p. 55],
and evidence for it is well-documented throughout his monograph.
17 Sandbach’s explanation of this term is as follows: “... Aristotle did write some works,
now lost, of which some were dialogues, intended for a wider public than the students who
were attached to his school. Later scholars, and probably Aristotle himself, referred to these as
‘exoteric’” [Sandbach, 1985, p. l].
406
Robert R. O’Toole and Raymond E. Jennings
of Stoic doctrines [Sandbach, 1985, p. 57]. Moreover, he quite forcefully expresses
the view that “it is a mistake to proceed on the a priori assumption that the
Stoics must have known the opinions expressed in his school-works, must have
understood his importance sub specie aeternitatis, and must therefore have been
influenced by him” [Sandbach, 1985, p. 57]. It is important to see that Sandbach
is not ruling out the possibility of Aristotelian or Peripatetic influence, but rather
urging a more careful and less biased approach to the question.
As might be expected, the foundations for many of the central tenets of Stoicism
can be discerned in the philosophical education of Zeno, and though it would be
interesting to trace the sources of the full range of Stoic doctrines, such pursuits
will be restricted in this introduction to considering those influences which would
seem likely to have affected the origins of Stoic logical theory. Probably Zeno’s
most significant contribution to Stoic logic is his work in epistemology.
It has been widely accepted among modern commentators that Zeno received
his education in logical techniques directly from the Megarians, and in particular,
from Diodorus Cronus (cf. Long [1986, p. Ill]; Mates [1953, p. 5]; Kneale and
Kneale [1962a, p. 113]; Rist [1978, p. 388]). An article by David Sedley, however,
would seem to raise some doubt that this assumption can be maintained. What
is called into question is not whether Zeno was indeed a student of Diodorus, for
this would seem to be beyond dispute, but rather whether Diodorus himself can
be established as belonging to the Megarian School.
The Megarian School was founded by Euclides, a pupil of Socrates (DL 2.47)
and a native of Megara on the Isthmus (DL 2.106). He was succeeded as head of
the school first by Ichthyas and later by Stilpo, also a native of Megara in Greece
(DL 2.113). Evidently, since Diodorus can trace his philosophical lineage back
to Euclides through Apollonius Cronus and Eubulides (DL 2.110-11), it has been
generally thought that he also was a member of the Megarian school; hence, the
Megarian connection with respect to the source of Zeno’s logical doctrines would
seem assured. Sedley, however, has presented what seems to us a convincing
argument to the effect that Diodorus belonged rather to a rival school which was
called the Dialectical School (Sedley, [1977, pp. 74-75]; cf. Sandbach, [1985, p.
18])-
At 2.106 Diogenes reports that the followers of Euclides were called Megarians
after his birthplace. Later they were called Eristics, and later still, Dialecticians.
Sedley argues for the possibility that these remarks should not be interpreted, as
they usually are, to mean that this was one and the same school known at different
times by different names, but rather that these names designated splinter groups
whose raisons d’etre were different enough from that of the Megarian School to
warrant viewing them as distinct schools [Sedley, 1977, p. 75]. According to Sed¬
ley, several sources inform us that the Dialecticians recognised Clinomachus of
Thurii, a pupil of Euclides, as the founder of their school [Sedley, 1977, p. 76].
However, since the name ‘Dialectician’ was first coined for the school by Dionysius
of Chalcedon (DL 2.106), an “approximate” contemporary of Diodorus (Sedley
[1977, p. 76]), it seems more likely not that Clinomachus actually founded the
The Megarians and the Stoics
407
school, but rather that he was recognised by its members as the source of the
ideas foremost in their teachings [Sedley, 1977, p. 76]. As Sedley points out, prac¬
tically nothing is known of Clinomachus’ doctrines, except for the fact, reported
by Diogenes Laertius (2.112), that he was the first to write about axiomata 18
(o^icopaxa) and predicates (xaxrjyoprjpaxoi). But this fact is of “utmost signifi¬
cance” [Sedley, 1977, p. 76], for both of these notions, as will become apparent
in the sequel, are fundamental constituents of the conceptual apparatus in Stoic
logic.
The Dialectician Diodorus and his pupil Philo apparently engaged in a famous
debate about the criterion according to which the consequent of a conditional
axioma (proposition) ‘follows’ from the antecedent (PH 110-12; AM 112-17). 19
The controversy between these logicians was apparently of interest not only to
themselves, for Sextus Empiricus reports that Challimachus, who served in mid-
third century B.C. as the chief librarian at the great library in Alexandria, wrote to
the effect that even the crows on the rooftops, having repeatedly heard the debate,
were cawing about the question which conditional axiom,ata are sound (AM 1.309-
10). The debate was extended by the introduction of two additional accounts, one
of which is attributed to Chrysippus on fairly strong evidence (e.g., Gould, [1970,
pp. 74-76]), but the other not attributed to any particular philosopher or even to
any particular school. Given Zeno’s association with Diodorus and Philo it seems
fairly certain he would have taken part in the debate; moreover, he wrote a book
On Signs (rtEpl cr][idu>v) (DL 7.4), and a ‘sign’ is defined by the Stoics as “the
antecedent axioma in a sound conditional, capable of revealing its consequent”
(AM 8.245). Hence he would have had an interest in the criterion for a sound
conditional axioma, perhaps even offering a view of his own. We will argue in the
sequel (see page 142) that given his motivation and particular interests in logic,
it is unlikely that he would have opted for the view of either Diodorus or Philo.
And since the third account can be attributed to Chrysippus with some certainty,
one may conjecture that Zeno supported the fourth view. Furthermore, if Long
and Sedley are correct, there may be no significant difference between the third
and fourth statements of the criterion [Long and Sedley, 1990, 1.211], so that
one might suppose, as long as we are in the realm of speculation, that the third
criterion represents a tighter version of the fourth account.
Evidently, then, there is some reason to believe that Zeno was influenced by
Diodorus and Philo through a familiarity with these ideas, and that it was the
Dialectic School rather than the Megarian School which was the important in¬
spiration for the development of Stoic logic (Sedley [1977, p. 76]). It may be,
nevertheless, that Stilpo played some part as an influence on Zeno’s logical edu¬
cation, and indeed, there may be some overlap in certain areas, for after all, if
what has been argued above is correct, the source of the logical doctrines of both
18 Axiomata are somewhat akin to propositions, but differ in some important ways. For a
discussion see section 6 on page 463.
19 For the details on this debate, see Section 8.1.
408
Robert R. O’Toole and Raymond E. Jennings
the Megarians and the Dialecticians is, for the most part, one and the same. 20
Given that Stilpo was one of those who came to listen to Diogenes the Cynic (DL
6.76), it is possible, as Sandbach submits [Sandbach, 1975, p. 22], that it was not
Stilpo’s logical tutoring but rather his moral teachings which attracted Zeno to
the Megarian School, for these instructions were probably not unlike those of the
Cynics. On the other hand, J.M. Rist puts forward the hypothesis that Zeno be¬
came dissatisfied with Cynic ethical doctrine and its rather circumscribed concept
of ‘life according to nature’, and so was looking to develop an account of nature
with a basis in physical theory—any such account being rejected by the Cynics
[Rist, 1978, pp. 387-88]. The difficulty in such an undertaking is summed up by
Rist as follows:
In Zeno’s time and before ... the problems confronting a philosopher
who has come to the conclusion that he must embark on the study of
nature ... is that Parmenides and his Eleatic successors had attempted
to rule out such study altogether, and before it could be taken up,
philosophers deemed it desirable to propose ways by which Parmenides’
ban could be overcome [Rist, 1978, p. 388].
Hence “for the would-be cpucuxoc the acquisition of a certain familiarity with
Eleatic procedures would be a sine qua non ” [Rist, 1978, p. 388]. According
to the testimony of Diogenes Laertius, Euclides “applied himself to the writings
of Parmenides” (2.106), and since Stilpo was a pupil of Euclides (or at least, of
one of Euclides students) (2.113), he would no doubt be familiar with the argu¬
ments of Parmenides; hence, it is possible that Zeno was attracted to him for this
reason. And certainly in the Stoic theory of a coherent and continuous universe
held together by a pervasive pneuma immanent in all matter, (e.g., Alexander de
mixtione 216.14-17) there is some evidence that Zeno took up Parmenides’ thesis
of the unity of being.
Another possibility, not necessarily an alternative, is that Zeno was attracted
by Stilpo’s fame in the posing of logical problems and fallacies. One account has
it that Zeno once paid twice the asking price for seven dialectical forms of the
‘Reaper Argument’, so great was his interest in such things (DL 7.25). Probably
of more reliability, we have Plutarch’s testimony that Zeno would spend time
solving sophisms and would encourage his pupils to take up dialectic because of its
capacity to assist in this endeavour (de Stoic repugn. 1034e). Furthermore, Stilpo
20 It should be noted in passing that it cannot be assumed that either Sextus Empiricus, or any
other late commentator or doxographer, in using the term ‘hoi dialektikoi ’, is referring specifically
to the Dialectical School. Sedley points out at least two reasons for supposing that this is so:
first, the term ‘ dialektikos ’, was commonly used to designate anyone who used the method of
argumentation from which the Dialectical School got its name, that is, the method of putting an
argument in the form of question and answer (75; cf. DL 2.106); moreover, to quote Sedley, “by
the time of Chrysippus, in the late third century, [dialektikos] is the standard term for ‘logician’”
[Sedley, 1977, p. 75], Now since the Stoics were recognised for their logical acumen above and
beyond any rival school, it seems more likely that when Sextus refers to ‘hoi dialektikoi ’, for the
most part, he means the Stoics.
The Megarians and the Stoics
409
is reported by Diogenes Laertius to have “excelled all the rest in the invention of
arguments and in sophistry” (DL 2.113). The story goes that during a banquet
at the court of Ptolemy Soter he addressed a dialectical question to Diodorus
Cronus which the latter was unable to solve. Because of this failure, Diodorus
was reproached by the king and subsequently received the derisive name ‘Cronus’.
This caused him so much anguish that he wrote a paper on this logical problem
and “ended his days in despondency” (DL 2.112). Even taking into account the
likelihood that this story might be somewhat apocryphal, it probably can be taken
as a reliable indicator of Stilpo’s skill as an inventor of logical arguments and
puzzles. As an aside, recalling Sedley’s argument cited above, it would also seem
to point to a certain tension between Stilpo and Diodorus, a tension that one
would not normally expect if they had been members of the same school.
Another logical doctrine in which Zeno may have been influenced by Stilpo
is that concerning his rejection of the ‘forms’ or ‘ideas’. According to Diogenes
Laertius, Stilpo used to demolish the forms or universals, saying that whoever
asserts the existence of Man refers to nothing, for he neither refers to this particular
man nor to that; hence, he refers to no individual man (DL 2.119). And according
to the testimony of Stobaeus. similar opinions were held by Zeno, for he and his
followers relegated such ‘ideas’ or ‘concepts’ (swo/paxa) to a sort of “metaphysical
limbo”, referring to them as ‘pseudo-somethings’ (tbaavet xtvcc) ( eclog . 1.136.21).
It should be noted that this stance does not represent a rejection of all those things
which we refer to as ‘universals’. Common nouns such as ‘man’ or ‘horse’ were
taken to refer to the essential quality - and all qualities are corporeal - which made
something either a man or a horse (DL 7.58). It is if we were to use the term ‘man’
to refer to the genetic material which differentiates us from other creatures.
Kleanthes (331-232 B.C.)
After the death of Zeno in 261/2 B.C., Kleanthes, a native of Assos on the Troad,
became head of the Stoic School (DL 7.168). According to the historian Antis-
thenes of Rhodes, Kleanthes was a boxer before taking up philosophy (DL 7.168).
Upon his arrival at Athens he fell in with Zeno and was introduced to Stoic teach¬
ings which, in spite of having no natural aptitude for physics and of being ex¬
tremely slow (DL 7.170), he studied “right nobly”, remaining faithful to the same
doctrines throughout (DL 7.168). Zeno compared him to those hard waxen tablets
which are difficult to write on but which retain well the characters written on them
(DL 7.37). Kleanthes was perhaps the most religious of the Stoics, as witness his
well-known Hymn to Zeus. 21 He was acclaimed for his industry and perseverance
(DL 7.168), for it was said that he came to Athens with only four drachmas in his
possession (DL 7.168), and so was forced to work drawing water for a gardener by
night in order to support himself while studying philosophy by day (DL 7.169).
21 The Greek text is in Stobaeus Eclogae Physicae et Ethicae, vol. 1, page 25, line 12 to page
27, line 4. The text is translated in Long and Sedley [1990, 1.326-327]. There is also a translation
in somewhat more archaic (poetical?) language in Sandbach [1975, pp. 110—111].
410
Robert R. O’Toole and Raymond E. Jennings
Apparently he never got far beyond this impecunious state, for the story goes that
he was too poor even to buy paper, and so used to copy Zeno’s lectures on oyster
shells and the blade-bones of oxen (DL 7.174). At some point, however, he must
have gained access to a supply of writing material, for Diogenes has compiled a
list of his writings which includes about sixty books.
Of these sixty books catalogued, a series of four is listed under the title Inter¬
pretations of Heraclitus (DL 7.174). A connection with Heraclitus is also indicated
by the report of Arius Didymus to the effect that Kleanthes, comparing the views
of Zeno with those of other natural philosophers, says that Zeno’s account of the
soul or psyche (c[>uxd) is similar to that of Heraclitus (fr. 12 DK; DDG 470.25-
471.5). Other than a book under the title Five Lectures on Heraclitus attributed
to the Stoic Sphaerus, a pupil of Kleanthes (DL 7.178), there is little other direct
evidence to support the hypothesis of the influence of Heraclitus on Stoicism and
on Zeno in particular. This hypothesis is expressed, for example, in the following
statement by A.A. Long: “Heraclitus’ assumption that it is one and the same logos
which determines patterns of thought and the structure of reality is perhaps the
most important single influence upon Stoic Philosophy” [Long, 1986, p. 131]. It
is also expressed by G.S. Kirk, but with reservations:
Although Zeno must have based his physical theories particularly upon
Heraclitus’ description of fire, he is never named in our sources as
having quoted Heraclitus by name; while Kleanthes evidently initiated
a detailed examination of Heraclitus with a view to the more careful
foundation of Stoic physics upon ancient authority. . . . and there is
reason to believe that he made some modification of Zeno’s system in
the light of his special knowledge of the earlier thinker ... [Kirk, 1962,
pp. 367-68].
It is not a straightforward matter to see what can be made out from these cir¬
cumstances. What seems likely is that Zeno tacitly appealed to Heraclitus in
formulating his views on physics and cosmology, and that it was left to Kleanthes
to make explicit this appeal, modifying the theory where it seemed appropriate to
do so.
We can also look to the catalogue of Kleanthes’ writings reported by Diogenes
Laertius for assistance in giving an account of Kleanthes’ contribution to Stoic
logical theory. There is a set of three books under the title nepi zov Aoyov (DL
7.175) which one might take to be about logic, especially since Hicks translates
this title as Of Logic. It seems to us, however, that Of the Logos would be equally
possible. In light of Kleanthes’ interest in Heraclitus, and in light of the apparent
debt—pointed out by Long in the passage above—which the Stoics owe to Heracli¬
tus for their concept of the logos , it seems less likely that these books of Kleanthes
are about logic than that they are about the logos. There are three other titles,
however, which would appear to be uncontroversially on logical topics. These are:
Of Dialectic, Of Moods or Tropoi, and Of Predicates (DL 7.175). As to the first
title, not much can be said, for there is little in the sources to indicate Kleanthes’
The Megarians and the Stoics
411
particular thoughts on dialectic as such. However, more can be said about the
subjects of the other two books, for it is of some interest that Kleanthes wrote
about them.
Concerning the book about tropoi , Galen reports that logicians ( dialektikoi )
call the schemata of arguments by the name ‘mode’ or ‘fropos’ (xpoitoc). For
example, for the argument which Chrysippus calls the first indemonstrable (6
xpuTOC avaTtoSci-xoc) and which we would call modus ponens, the mode or tropos
on the Stoic account is as follows: If the first, the second; but the first; therefore,
the second ( inst. log. 15.7). Now according to Galen, since the major premisses
(jtpoTtiaEit;) 22 iv cKjtXXoYtcrpc; ocp xrju; aopx (tv xqtc; <;ctae, xrje covSixiovaX xpeptat;) ape
SexepfiLvaxte (r)yepovixaf) of the minor premisses (xpooXfjtJien;), Chrysippus and his
followers call such a proposition or axioma not only determinative but also tropic
(xporaxov). 23 What is of interest here is that Kleanthes’ concern with tropoi may
be an indication that he had some knowledge of the so-called indemonstrables,
that is, the five argument schemata which Chrysippus took as the basis of the
Stoic theory of inference. 24 As to the book about predicates (xaxr)yopf]paxa), we
also have a passage of Clement of Alexandria which maintains that Kleanthes
and Archedemes called predicates lekta (Xcxxa) (strom. 8.9.26.3-4). According to
Michael Frede, this testimony indicates that Kleanthes was the first philosopher
to use the term ‘lekton ’ ([Frede, 1987b], 344). This innovation is quite significant,
for the concept of the lekton is well recognised as possibly the most fundamental
notion in Stoic semantic theory. Frede suggests, however, that this passage is
evidence that the concept may have been introduced by the Stoics in the ontology
of their causal theory rather than in their philosophy of language ([Frede, 1987a,
p. 137]; cf. Long and Sedley [1990, 2.333]). At any rate, one might conjecture
that Kleanthes had some hand in the development of this concept.
We have the testimony of Epictetus of a book on a logical topic written by
Kleanthes but not recorded by Diogenes. Although he does not give its name, this
work, according to Epictetus, was on the so-called ‘Master Argument’ of Diodorus
Cronus (disc. 2.19.9). The Master Argument was apparently posed by Diodorus
in order to establish his definition of the possible (disc. 2.19.2; cf. Alexander in
22 The term protasis (KpoTaaic) is used in Sextus Empiricus to refer to the major premiss
of a categorical syllogism (PH 2.164; 195). Galen seems here to be extending the use of this
term to refer to the major premiss of a Stoic hypothetical syllogism as well. Thus it would be
interchangeable with the term lemma (Afjppa), which, according to Diogenes Laertius, the Stoics
used to refer to the major premiss of an argument (DL 7.76).
23 In his commentary on this section of the Institutio, Kieffer provides the following explanation:
“Galen’s point in calling the major premiss in a hypothetical syllogism determinative of the
minor is that the minor premiss is either one of the members of the hypothetical major or its
contradictory” ([Kieffer, 1964], 92). Thus in the case where the major premiss of the hypothetical
syllogism is a conditional, the minor premiss will be either the antecedent of the conditional (as in
modus ponens) or the negation of the consequent (as in modus tollens ). Note that ‘hypothetical
syllogism’ here covers any syllogism which is not categorical, and for the Stoics this includes not
only syllogisms with a conditional as major premiss but also those with either a disjunctive or a
negated conjunction.
24 For a more complete account of the Stoic argument schemata called the ‘indemonstrables’,
see page 474.
412
Robert R. O’Toole and Raymond E. Jennings
an. pr. 184.5), this definition being ‘The possible is that which either is or will
be’ (6 f) scttiv rj earai. (in an. pr. 184.1). 25 A definition of the possible attributed
to the Stoics both by Diogenes Laertius (7.75) and by Boethius (in de interp.
234.27) is that the possible is ‘that which admits of being true and which is not
prevented by external factors from being true’ (DL 7.75). Now both Kleanthes
and Chrysippus rejected Diodorus’ interpretation of the Master Argument (Epict.
disc. 2.19.6) and presumably, therefore, would have also rejected his account of
the possible, and since the Stoic characterisation given by Diogenes and Boethius
is not attributed to any specific Stoic, it is open to debate whether to credit it
to Chrysippus or to Kleanthes. There is, however, a passage in Plutarch which
would seem to indicate that Chrysippus warrants the attribution (de Stoic repugn.
1055d-e). However that may be, it is evident that Kleanthes had an interest in
questions about modality and no doubt gave some account of the possible and the
necessary.
Chrysippus (circa 282-206 B.C.)
Chrysippus of Soli 26 succeeded Kleanthes in 232 B.C. to become the third leader of
the Stoic School (DL 7.168; 1.15-16). There is not much information on his early
life. Hecato, the Stoic, says that he came to philosophy because the property he
had inherited from his father had been confiscated by the king (DL 7.181). And
there is a story that he was a long distance runner before taking up philosophy
as a pupil of Kleanthes (DL 7.179). Even as a student he seemed to possess a
good deal of confidence in his abilities, especially in logic, for he used to say to
Kleanthes that he needed to be instructed only in the doctrines; the proofs he
would discover himself (DL 7.179). His relationship with Kleanthes was somewhat
troublesome to him at times. On the one hand, he showed a great deal of respect
for Kleanthes, deflecting to himself the attacks of certain presumptuous dialecti¬
cians who would attempt to confound Kleanthes with their sophistical arguments.
Chrysippus would reproach them not to bother their elders with such quibbles,
but to direct them to his juniors (DL 7.182). On the other hand, he himself would
sometimes contend with Kleanthes, and whenever he had done so, would suffer a
good deal of remorse (DL 7.179).
Chrysippus apparently left the Stoic school while Kleanthes was still alive, be¬
coming a philosopher of some reputation in his own right (DL 7.179), and on the
authority of the historian Sotion of Alexandria, Diogenes tells us that he also stud¬
ied philosophy for some period under Arcesilaus and Lacydes at the Academy (DL
2S cf. Boethius in de interp. 234.22-26: quod aut est aut erit. As Benson Mates points out,
Boethius in this passage also gives definitions of the related terms ‘impossible’, ‘necessary’, and
‘non-necessary’, and based on the construction of these other definitions, one can conjecture that
the above definition of the possible was “slightly elliptical.” The full definition should have been
‘that which is or will be true' 1 [Mates, 1953, p. 37],
26 According to Zeller, the view of most writers was that Chrysippus was born at Soli in Cilicia;
however, since his father emigrated to Soli from Tarsus, it is possible that Chrysippus was born
there instead ([Zeller, 1962], 45n5).
The Megarians and the Stoics
413
7.183). This would explain, according to Diogenes, his arguing at one time for
common experience (auvrjfkiot), 2 ' avS avoxrjEp xipe aycuvax ix (AA 7.184). BtJj xrjtc;
pepapx AioyEvec; u; vo 8ou|3x pstpepptvy xo xrj£ <pacx xrjax ppcJiatTcituc; topoxE a aEpicc;
ocp cul; poox<; uvSsp xr]E xixXe Ayaivax oppov Elfnepiev^e (Kaxa xrjc auvrjDeLCfc;), as
well as a series of seven books under the title In Defence of Common Experience
(Yttcp xfj<; auvrydeiac) (DL 7. 198). In this regard, according to Cicero, some later
Stoics complained against him for providing Carneades and the Academy with
arguments with which to assail against the whole of common experience, as well
as against the senses and their clarity and against reason itself ( acad . 2.87; cf.
Plutarch de Stoic repugn. 1036b-c).
Chrysippus was an extremely prolific writer. Diogenes Laertius reports that in
all he wrote seven hundred and five books (DL 7.180), of which three hundred and
eleven were on logic (DL 7.198). And Diogenes provides an inventory of about
three hundred and seventy five of them, the majority of these being books on logic
(DL 7.189-202). Diogenes also cites the testimony of Diodes Magnes who claims
that Chrysippus wrote about five hundred lines a day (DL 7.181). It would seem,
however, that in the opinion of many, such a profusion of material did not come
without a price, for the ancients, according to Zeller, were unanimous in putting
forward a litany of complaints against the literary style of these texts [Zeller, 1962,
pp. 47-48]. However, this criticism is somewhat mitigated by Zeller’s comment
that “with such an extraordinary literary fertility, it will be easily understood that
their artistic value does not keep pace” [Zeller, 1962, p. 47]. But whatever are the
merits of these criticisms, one cannot help but speculate on how different would
have been our understanding of Hellenistic philosophy had even a few of these
works survived.
With the death of Kleanthes, Chrysippus returned to Stoicism to become leader
of the school. In that capacity he was, “in the opinion of the ancients, ... the
second founder of Stoicism” Zeller, [1962, p. 45], for it was said, according to Dio¬
genes Laertius, that “if there had been no Chrysippus, there would have been no
Stoa” (DL 7.183). Gould takes this to refer to the belief that Chrysippus “revived
the Stoa after the crushing blows dealt it by Arcesilaus and other Academics”
[Gould, 1970, p. 9]. He continues:
In antiquity, then, even outside the school, Chrysippus was regarded
as an eminently capable philosopher, as an extraordinarily skilful di¬
alectician, and as one who came to the defense of the Stoa in a crucial
moment, namely, when it was about to encounter its death blow from
a rival school in Athens, the Academy, which had then become the
stronghold of scepticism [Gould, 1970, p. 9].
27 Here, and in Plutarch’s De Stoicorum repugnantiis at 1036c, as well as in Epictetus’ The
Discourses at 1.27.15-21, the term ‘ouvriDeia’ seems to have the meaning ‘common experience’;
on the other hand, in Diogenes Laertius at 7.59 it would appear to mean ‘ordinary language’,
and in De Stoicorum repugnantiis at 1048a, to mean ‘common use of language’ (cf. the entry in
Liddel and Scott, 11.2).
414
Robert R. O’Toole and Raymond E. Jennings
It was no doubt his skill as a dialectician which enabled Chrysippus to defend
so well the doctrines of the Stoa, for he was considered by many to have been
one of the foremost logicians of Hellenic Greece. In fact, according to Diogenes
Laertius, he was so renowned for his logical acumen “that it seemed to most
people, if dialectic was possessed by the gods, it would be none other than that of
Chrysippus” (DL 7.180). Perhaps there would be no more fitting way to conclude
this section on Chrysippus than to quote the words of Long and Sedley in their
source book The Hellenistic Philosophers: “In the period roughly from 232 to
206 [Chrysippus] was to ... develop all aspects of Stoic theory with such flair,
precision and comprehensiveness that ‘early Stoicism’ means for us, in effect, the
philosophy of Chrysippus” [Long and Sedley, 1990, 1.3].
3 PRELIMINARIES
Several of our sources attest to a tripartite division of philosophy by the Sto¬
ics. These branches are logic, physics, and ethics (DL 7.39; Aetius plac. DDG
273.11; Plutarch de Stoic repugn. 1035a). According to Diogenes Laertius, Zeno
of Citium, in his book On Discourse (Kepi A oyov), was the first of the Stoics to
make this division (DL 7.39). Diogenes also informs us that Zeno arranged these
topics with logic first, physics second, and ethics third (DL 7.40), although it is
somewhat unclear whether the standard for this arrangement is according to in¬
trinsic importance or according to teaching priorities. Perhaps what it reflects is
the relationship between these parts as it is represented in one of the many similes
that the Stoics drew upon for illustration. Their philosophical system, they said,
is like a fertile field with logic as the surrounding wall, physics as the soil or trees,
and ethics as the fruit (DL 7.40). It is clear on this conception that logic is given
the task of protecting the system from external threats—the first line of defence,
as it were, and the aspect of logic emphasised is skill in dialectic in the sense of
mustering counter-arguments and solving sophisms (cf. Plutarch de Stoic repugn.
1034e). But there is another aspect of logic in which the sense of dialectic stressed
is that in which it signifies “the testing of hypotheses and the quest for ultimate
principles or true definitions, which are the essential procedures of every meta¬
physician” (Long and Sedley 1.189). In The Discourses , Epictetus surmises that
the reason why the philosophers of the Old Stoa put logic first in the exposition of
their doctrine is that it is in the study of logic that one comes to understand the
criterion by which one judges in other pursuits what is true. So it is, according to
Epictetus, that:
... in the measuring of grain we put first the examination of the mea¬
sure. And if we should neither first define what a modius is, nor first
define what a scale is, how shall we be able to measure or weigh any¬
thing? So with the subject of logic, how shall we be able to investigate
accurately and understand thoroughly anything of other subjects if we
neither thoroughly understand nor accurately investigate that which is
The Megarians and the Stoics
415
the criterion of other subjects and that through which other subjects
are thoroughly understood? (disc. 1.17.7-8)
In the first part of this section, we explore the hypothesis that the development of
the Stoic system as it is discernible in the philosophical education of Zeno followed
the reverse order to that envisaged above: it evolved from a primary interest in
ethics and thence to physics and logic. And the aspect of logic cultivated in this
succession is that characterised both in the quotation from Long and Sedley and
in the quotation from The Discourses: that is, logic as concerned with truth,
knowledge, definitions, and other elements of reason, and, as Zeno says, with
understanding “what sort of thing each of them is, how they fit together and
what their consequences are” (Epictetus disc. 4.8.12). Given this understanding
of Zeno’s development of the Stoic system, we shall suggest, in the second part of
this section, an interpretation of Stoic logical doctrines which may be perceived as
being motivated by this evolution, doctrines propounded either by Zeno himself
or by his successors.
3.1 Stoic Ethics: the Motivational Basis
From Crates and the Cynic School Zeno doubtless inherited the foundation for
his moral theories. But the Cynics apparently devoted themselves only to ethics,
choosing to do away with the topics of logic and physics (DL 6.103). Zeno, on
the other hand, clearly did not reject these topics, for it is uncontroversial that he
laid the foundations not only for Stoic ethics, but for physics and logic as well. It
would seem evident, therefore, that at some point he broke ranks with the Cynics,
choosing a philosophical curriculum richer than that of his mentors. 28 The point
on which Zeno diverged from the Cynic path is the premiss described by Long and
Sedley as “the bastion of Stoic ethics,” that is, “the thesis that virtue and vice
respectively are the sole constituents of happiness and unhappiness” [Long and
Sedley, 1990, 1.357]. 29 AcpopSivy xo xr)it; ieo>, aope xrpya—cpop e^apTtXe, rjeaXxr),
weaXxr), Peauxcjj, av§ TuyjiaicaX axpevyxr)—cav vsixr]£p pevetpix (obcpeXa) nor harm
28 This thesis is put forward by J.M. Rist in his essay “Cynicism and Stoicism” which appears
as Chapter 4 of his Stoic Philosophy.
29 The terms which Long and Sedley render as ‘happiness’ and ‘unhappiness’ are ‘eOSaipovia ’
(eudaimonia ) and ‘xaxoSaipovia ’ ( kakodaimonia ) respectively. Now although these are standard
translations, it has been suggested that they fail to capture the notion which they are intended
to express. Sandbach has the following to say on this point:
eudaimonia , although something experienced by the man who is eudaimon , is (per¬
haps primarily) something objective, that others can recognise—having a good lot
in life. ... Thus the Stoics did not attempt to describe eudaimonia as a subjective
feeling, but identified it with such things as ‘living a good life’, ‘being virtuous’,
or ‘good calculation in the choice of things that possess value’ ... For the Stoic,
who confines the word ‘good’ to the morally good, it is consistent that a good life
is a morally good life and the well-being indicated by eudaimonia is unaffected by
what is morally indifferent, however acceptable [Sandbach, 1975, p. 41].
In another place Long himself renders eudaimonia as ‘self-fulfilment’ ([Long, 1971], 104), a
translation which probably better comprehends the meaning, although still not completely.
416
Robert R. O’Toole and Raymond E. Jennings
(pX&7iT£i): they are neither necessary nor sufficient for a virtuous—which is to say
a moral —life, and thus can have no effect on one’s eudaimonia ; hence, they are
called ‘indifferent’ (axiotcpopoc;) (DL 7.102-03). This thesis was undoubtedly an
endowment to Zeno from the Cynics, for we learn from Diogenes Laertius that
for the Cynics virtue is sufficient in itself to secure happiness (DL 6.11), so that
“the end for man is to live according to virtue” (teXoc etvai to xax' apexr)v £rjv),
a credo that was echoed by the Stoics (DL 6.104; cf. Stobaeus eclog. 2.77.9).
We also learn from Diogenes that the Cynics count as ‘indifferent’ whatever is
intermediate between virtue and vice (DL 6.105).
The Cynic stance toward the ‘indifferents’ can perhaps best be understood
by considering the views of Ariston of Chios, a pupil of Zeno’s whom we might
call a ‘neo-Cynic’, since he “greatly simplified Stoicism, so that it was hardly
distinguishable from the attitude of the Cynics” (Sandbach [1975, p. 39]; cf. Rist
[1969c, pp. 74-80]). Ariston recommended complete indifference to everything
between virtue and vice, recognising no distinctions among them and treating
them all the same, for the wise man, according to Ariston, “is like a good actor,
who can play the part of both Thersites and Agamemnon, acting appropriately in
each case” (DL 7.160). A problem for this credo is summarised by Cicero, who is
setting out the opinions of the Stoic Cato:
If we maintained that all things were absolutely indifferent, the whole of
life would be thrown into confusion, as it is by Aristo, and no function
or task could be found for wisdom, since there would be absolutely
no distinction between the things that pertain to the conduct of life,
and no choice need be exercised among them (de fin. 3.50; trans.
Rackham).
This criticism, which was no doubt a standard reproach among the ancients, might
be expressed by the observation that “[Aristo’s] position robbed virtue of content”
[Sandbach, 1975, p. 38]. It is unclear whether Zeno himself subscribed to this
assessment; however, it would seem likely that he did. He was, after all, in at¬
tendance at Polemo’s lectures, and there he would no doubt have become familiar
with the view, ascribed by Cicero to Xenocrates as well as to his followers, that
the ‘end of goods’ (finis bonorum ) is not limited to virtue alone, but includes just
those things which belong to the class which the Cynics and Ariston held to be
‘indifferents’ (de fin. 4.49; Tusc. disp. 5.29-30). 30
At any rate, although he may have agreed with this critique of Cynic views
to the extent of granting that such things as “health, strength, riches, and fame”
(de fin. 4.49) have some value, Zeno was nevertheless unwilling to depart from
Cynic tenets to the point of admitting that any of the indifferents were required
for eudaimonia. His solution to this dilemma was to introduce a classification
30 The doctrine of the Academy would appear to go back to Plato himself, for in Laws 661a-d,
he has the Athenian say that things such as health, beauty, wealth, and acute sensibility all are
to be counted as goods, but only in the possession of the just and virtuous; in the possession of
one who is not so, however, all these things are rather evils than goods.
The Megarians and the Stoics
417
of the indifferents distinguishing those which are ‘according to nature’ (xdt xaxa
cpuatv), those which are ‘contrary to nature’ (xa rapa cpuaiv), and those which are
neither (Stobaeus eclog. 2.79.18). Indifferents which are according to nature ( ta
kata physin ) are such things as health, strength, sound sense faculties, and the like
(eclog. 2.79.20). All indifferents which are kata physin have ‘value’ (ot$foi), whereas
those contrary to nature have ‘disvalue’ (dmdfta) (eclog. 2.83.10). A categorisa¬
tion of indifferents which would seem to coincide with this division is that which
distinguishes them according to those which are preferred’ (xa xporjYpeva), those
‘rejected’ (xd &jt07tpor)ypeva), and those neither preferred nor rejected (DL 7.105;
Cicero de fin. 3.51; Stobaeus eclog. 2.84.18). According to Diogenes Laertius, the
Stoics teach that those indifferents which are preferred (ta proegmena) have value
(axia), whereas those rejected have ‘disvalue’ (apaxia) (7.105). Thus it would ap¬
pear that the class of those indifferents which are L kata physin' is coextensive with
l ta proegmena'.
Value is defined by the Stoics as having three senses, only two of which are
relevant in this context. 31 Foremost, it is the property of contributing to a harmo¬
nious life, this being a characteristic of every good (dycrfld) (DL 7.105). However,
since no goods are among the preferred (Stobaeus eclog. 2.85.3), this connotation
of axia must refer only to goods and not to the preferred, and hence designates
value in an absolute sense; indeed, the things which have value in this sense are
called ‘xipf]v xocfl' coho’ (eclog. 2.83.12), which may be rendered as ‘value per se ’
(cf. Cicero de fin. 3.39 and 3.34). The second sense is that of some faculty or use
which contributes indirectly (pear]) to life according to nature (DL 7.105). Things
which have value according to this sense of axia are ‘selected’ (exXexxixoc;), in
Antipater’s phrase, on which account, when circumstances permit we choose these
particular things rather than those: for example, health against illness, life against
death, and wealth against poverty (Stobaeus eclog. 2.83.13). The notion of value
among those indifferents having ‘preferred’ status, as well as the responsibility of
the moral agent with respect to these notions, is well summarised by Long and
Sedley in the following passage:
This ‘selective value’, though conditional upon circumstances (con¬
trast the absolute value of virtue), resides in the natural preferability
of health to sickness etc. That is to say, the value of health is not
based upon an individual’s judgement but is a feature of the world.
The role of moral judgement is to decide whether, given the objective
preferability of health to sickness, it is right to make that difference
the paramount consideration in determining what one should do in
the light of all the circumstances. In the case of those indifferents of
‘preferred’ status, there will be ‘preferential’ reason for selecting these
‘when circumstances permit’. It is up to the moral agent to decide,
from knowledge of his situation, whether to choose actions that may
31 In the third sense, according to Diogenes, “value is the worth set by an appraiser, which
should be fixed according to experience of the facts, as, for example, wheat is said to be exchanged
for barley plus a mule (DL 7.105).
418
Robert R. O’Toole and Raymond E. Jennings
put his health at risk rather than preserve it, but the correctness of
sometimes deciding in favour of the former does not negate the normal
preferability of the latter [Long and Sedley, 1990, 1.358].
The hypothesis of ta kata physin as a sub-class of the indifferents, then, will
permit Zeno to maintain the thesis that only virtue is good, and at the same time
provides a content for his ethics. A concomitant result of this hypothesis will be
to lead to the reinstatement of physics and logic as legitimate components of a
philosophical education. The theory may be represented as positing three levels of
maturation in the moral agent (Cicero de fin. 3.20-21; 4.39; Aulus Gellius 12.5), 32
so that ta kata physin contribute content for Zeno’s theory in accordance with
the level of moral development of the agent (cf. Edelstein and Kidd, 155-57). In
each stage of development ta kata physin are associated with a category of acts
to which Zeno has given the name ‘appropriate acts’ (to; xctDyjxovxc;) (DL 7.108).
At the first level, instanced by babies and young children in whom the faculty of
reason or logos (Xoyoc) has not yet evolved, the agent is concerned with ‘primary’
things according to nature (rot upcoTot xorca tpuctv) (Stobaeus eclog. 2.80.7; Aulus
Gellius 12.5.7). These are the things toward which natural impulse (opprj) inclines
us in order to preserve and enhance our own constitution (Cicero de fin. 3.16;
Seneca epist. 121.14). Hence at this level an appropriate act would be just to
carry out those desires which impulse urges that we do; moreover, such acts would
entail no consequences for morality, since moral choice and responsibility requires
rationality.
At the second stage, subsequent to the emergence of the faculty of reason in the
agent, ta kata physin are the base ( proficiscantur ah initiis naturae ) (Cicero de
fin. 3.22) and the impetus or arche (apxh) (Plutarch comm. not. 1069e) for those
acts which reason convinces us to do, or those for which, when done, a reasonable
justification can be given (DL 7.108; Stobaeus eclog. 2.85.14). Ta kata physin are
themselves still the objects of appropriate acts, but now it is logos rather than
horme (impulse) which is active in the agent, directing his choices (Cicero de fin.
3.20). Concomitantly with the emergence of logos, comes the capability to form
‘conceptions’ or ennoiai (evvoioci). The gradual accumulation of the appropriate
stock of ennoiai will, in the end, endow the agent with the capacity to discern the
order and harmony in nature and to act in accordance with them (de fin. 3.21).
32 Long and Sedley note that in De finibus 3.17 and 20-21, Cicero “envisages five progressive
stages, each of which is represented as performance of ‘proper functions’ as these could evolve
for a human being” [Long and Sedley, 1990, 1.368]. Edelstien and Kidd, on the other hand,
stresses De finibus 4.39 where Cicero gives a threefold division of ta kata physin (Posidonius I:
the Fragments 155). It seems to us that there is a clear relationship between this division of ta
kata physin and the five stages of ta kathekonta as they are summarised at de fin. 3.20, wherein
the first two stages of ta kathekonta, i.e. “to preserve oneself in one’s natural constitution”
and “to retain those things which are in accordance with nature and to repel those that are
the contrary,” are associated with the first division of ta kata physin-, the second two stages,
i.e. “choice conditioned by reasoned action” and “such choice becoming a fixed habit,” with the
second division; and the last stage, i.e. “choice fully rationalised and in accordance with nature,”
with the third division. Hence in our presentation we have exploited this relationship by merging
these categorisations into one three-fold differentiation.
The Megarians and the Stoics
419
Having reached this state, the agent is at the threshold of the third stage of
moral development. At this third level, ta kata physin are no longer the impetus
for choice, but are merely the ‘material’ or hyle (0Xf|) of ta kathekonta (Plutarch
comm. not. 1069e; cf. Cicero de fin. 3.22-23 and Galen SVF 3.61.18); moreover,
the object of ta kathekonta at this level is not the attainment of ta kata physin ,
but rather wisdom of choice, which is, in effect, choice in accordance with virtue
(de fin. 3.22); hence, it is the moral character of the agent that determines the
appropriateness of the act (de fin. 3.59; Sextus Empiricus AM 11.200; Clement
SVF 3.515). Thus, even though there may be “a region of appropriate action
which is common to the wise and unwise” (Cicero de fin. 3.59), the appropriate
acts of the wise man, unlike those of the unenlightened, are consistently motivated
by reason. Appropriate acts at this stage are ‘perfect’ and are referred to as ‘right
actions’ (xaxopOupcKTa), since they contain all that is required for virtue (de fin.
3.24; Stobaeus eclog. 2.93.14).
It is evident, given the above characterisation of a ‘perfect’ kathekon or ‘right
action’ (katorthoma), that the Stoic sage will be the only one to reach the third
level of moral development, for only the wise man performs right actions (Cicero
de fin. 4.15). It would seem, therefore, that a function or task for wisdom, the
lack of which was seen as a shortcoming of the Cynic view as represented by
Ariston (see page 416), is found in the choices the wise man makes among ta kata
physin. Where this Stoic version of moral preference differs from what might be
called the ‘common sense’ account is that the object of such choices is not the
attainment of particular things which accord with nature—things such as health,
strength, wealth, and so on—but rather the attainment of a virtuous disposition
which functions consistently in the making of such choices (Cicero de fin. 3.22;
3.32). Thus, even though this feature of the Stoic position was much maligned in
ancient times (e.g., de fin. 4.46-48; Plutarch comm. not. 1060e), there is no doubt
that Zeno’s innovation provided the improvement he desired over Cynic doctrine
(Cicero de fin. 4.43).
A related shortcoming of the Cynic view is the doctrine to the effect that “virtue
needs nothing except the strength of a Socrates” and that “virtue is concerned
with deeds, requiring neither a host of rules nor education” (DL 6.11). One might
suspect that for Zeno the difficulty with this credo would have been the problem
that it rendered the content of virtue as quite arbitrary, dependent only on the
will of the wise man; 33 moreover, one might also surmise that Zeno would have
questioned how the ordinary person, not endowed by nature with the strength of a
Socrates, would be supposed to go about improving himself with respect to moral
rectitude. This difficulty was evidently addressed by the thesis that virtue is the
outcome of a developmental process. The latter premiss suggests that it should
33 So Sandbach: “What Zeno was probably afraid of was that what might be dignified with
the name of acts of will might in fact be acts of whim and caprice. Since virtue itself seemed so
difficult to understand or describe, the danger of this was very real indeed. That is why so many
of the Cynics give the impression of being merely irresponsible exhibitionists” [Sandbach, 1975,
p. 71],
420
Robert R. O’Toole and Raymond E. Jennings
be possible to learn to be virtuous, and, indeed, the idea that virtue is or can be
taught is explicitly reported as Stoic doctrine in several places (DL 7.91; Clement
SVF 3.225). Now although Zeno is not mentioned as advocating this view, both
Kleanthes and Chrysippus are; hence, there would seem to be no good reason not
to attribute it to Zeno as well. The details of the Stoic account of how virtue is
learned may be inferred from the sources; 34 what is relevant here, however, is the
contrast between this thesis and the preceding description of Cynic doctrine.
Given the account of moral development outlined above, it would seem that
Zeno would have been in a position to augment the Cynic dictum reported by
Diogenes Laertius that ‘the telos for man is to live in accordance with virtue’ (see
page 416). The right actions or katorthomata performed by the wise man are
mentioned in several places by Stobaeus as being actions performed according to
‘right reason’ (opfloc Xoyoc) ( eclog. 2.66.19; 93.14; 96.18); moreover, right reason
is described by numerous sources as being equivalent to virtue (Plutarch de virt.
mor. 441c; Cicero Tusc. disp. 4.34; Seneca epist. 76.11). Hence, the virtue of the
wise man consists in the perfection of his rationality with respect to the choices
he makes among ta kata physin (Seneca epist. 76.10; Cicero de fin. 3.22). These
choices are made in accordance with his own rational nature and in accordance
with the logos of the universe, the rationality of which he shares (DL 7.87-88;
Cicero de nat. deorum 1.36-39; 2.78; 133; 154; Seneca epist. 124.13-14).
According to Diogenes Laertius, Zeno’s position concerning the summum bonum
was that “the telos for man is to live harmoniously with nature” (teXoc eute to
6|ioXoyoup£wo<; rfj (puaa) (DL 7.87); Stobaeus, however, reports that this formu¬
lation is due to Kleanthes, whereas Zeno’s statement is simply that the telos is
“to live harmoniously” (to opoXoyoupevwc £fjv). This means to live in accordance
with a single harmonious logos, since those who live in conflict with this are not
eudaimones (eclog. 2.75.6-76.6). Chrysippus’, on the other hand, said that the
telos is “to live in accordance with experience of what happens by nature” (Cfjv
xar' epuciplav tcov cpuaei aupPaivovTwv) (eclog. 2.76.6-8). There would seem to be
no good reason to suppose that these definitions are incompatible in any way, for
according to Stobaeus, the augmentations to Zeno’s statement were proposed not
because Kleanthes and Chrysippus disagreed with him, but rather because they
assumed that his formulation was an ‘incomplete predicate’ (eclog. 2.76.2-3) and
they wished to make it clearer (eclog. 2.76.7). 35
Cicero reports a further statement which the Stoics declared to be equivalent to
Zeno’s representation. It asserts that the telos is “to live in the light of a knowl¬
edge of the natural sequence of causation” (vivere adhibentem scientiam earurn
rerum quae natura evenirent) (de fin. 4.14). The justification for this equiva¬
lence can evidently be inferred from several passages in the sources. First, the
Stoics called the natural sequence of causation ‘ heimarmene ’ (dpappevr)), usually
translated as ‘fate’ or ‘destiny’ (Cicero de div. 1.125-26; Aulus Gellius 7.2.3). In
addition, Stobaeus reports that heimarmene is the logos, or rational principle of
34 E.g., Cicero de fin. 3.33; Seneca epist. 120.4. See also Long [1986], 199-205.
35 See Sandbach’s discussion in The Stoics, [Sandbach, 1975, pp. 53-55].
The Megarians and the Stoics
421
the universe ( kosmos). It is “the logos according to which past events have hap¬
pened, present events are happening, and future events will happen” (Xoyoc xafi'
ov xa [rev yeyovxoxa yeyove, xa yivopeva ytvexai, xa 8e ysvqaopeva yevrjaexai.);
furthermore, Stobaeus informs us that the rational principle, in addition to be¬
ing called the logos, is also referred to as ‘truth’ (aXfpSeia), ‘explanation’ (odxlot),
‘nature’ (fiuau;), or ‘necessity’ (dcvdyxr)) ( eclog. 1.79.1-12; cf. Alexander de fato
192.25). The identification of heimarmene with logos , the rational principle of the
kosmos , and the fact that this principle is also referred to as physis , would seem
to establish the basis for the equivalence in question.
Thus Zeno’s interpretation of the Cynic doctrine that the end for man is to live
according to virtue can be formulated first by the statement that the telos is ‘to live
according to right reason’, since virtue and right reason are taken to be equivalent.
What this means, given the account of the development of the virtuous man, is
that the end for man is ‘to live in accordance with his own rational nature and in
accordance with the logos of the universe’. This latter version can be summarised
in turn by the statement that the telos for man is ‘to live harmoniously with
nature’, or equivalently, ‘to live in the light of a knowledge of the natural sequence
of causation’.
Evidently, if someone thought that to be a wise man one ought to live according
to nature in the sense of living according to a knowledge of the natural sequence
of causation, then he most likely would also think that the study of physics and
logic would be a requirement of a philosophical education. It is quite probable,
therefore, that Zeno’s concept of the telos would have led him to adopt a philosophy
in which physics and logic were as much a part of the curriculum as was ethics. The
discussion in the last few paragraphs would seem to show that the development
of Zeno’s notion of the telos is a result of his doctrine of ta kata physin. This
doctrine represented a major break with his Cynic roots inasmuch as it required
that some things which the Cynics had classified as absolutely morally ‘indifferent’,
be classified instead as ‘preferred’, in the sense that they are ‘according to nature’.
Hence it would seem not only that Zeno’s notion of ta kata physin itself represented
an important break with Cynic doctrine, but also that this notion led to a further
breach inasmuch as it induced him to include physics and logic in his philosophical
curriculum, contrary to the Cynics.
If the philosopher is to be educated in the study of physics and logic, the ques¬
tion arises concerning the scope and content of his knowledge in these subjects. A
passage from Seneca and one from Diogenes Laertius will be helpful here. Seneca
tells us that “the wise man investigates and learns the causes of natural phe¬
nomena, while the mathematician follows up and computes their numbers and
their measurements” (epist. 88.26) In a passage with a similar theme, Diogenes
Laertius tells us that the part of physics concerned with causation is itself divided
between the investigation of such things as the hegemonikon (qyepovixov)—that
is, the ‘leading part’ of the soul or psyche ((jjuyiQ), of what happens in the psyche,
of generative principles, and of other things of this sort. This is the province of
the philosopher. On the other hand, the mathematician is concerned with such
422
Robert R. O’Toole and Raymond E. Jennings
things as the explanation of vision, the cause of an image in a mirror, the origins
of weather phenomena, and similar things (DL 7.133). These passages would seem
to suggest that scope of the wise man’s knowledge of physics would probably not
be of particular data, but rather of general principles. 36
As to the question concerning the nature of the logical theory that the philoso¬
pher would need, we take it that the motivation for such a theory would be the
requirements of the ethical doctrine outlined above. 37 Thus, given that the wise
man’s aim is to live in harmony with a knowledge of the natural sequence of
causation, he will need to make correct judgements about the relations between
particular states of affairs, based on his knowledge of the general principles gov¬
erning such connections; moreover, he will need to comprehend the patterns of
inference which will allow him to effect such judgements. Evidently, these general
principles will be manifestations of the universal logos , and as such, given that
logos is another name for ananke (necessity) (Stobaeus eclog. 2.79.1-12; Alexan¬
der de fato 192.25), they will be embodied by necessary connections in nature;
moreover, one can surmise that the patterns of inference which emerge will consist
in part of representations of such necessary connections.
3.2 Inference and Akolouthia
At 7.62, Diogenes Laertius reports that, according to Chrysippus, dialectic is
about ‘that which signifies’ and ‘that which is signified’ (Kepi xct oqpalvovTa xal xa
orjpaivopeva). It is clear from the context that ‘that which signifies’ is a meaningful
utterance. Presumably, then, ‘that which is signified’ is the significatum of such
an utterance. Diogenes does not elaborate on what belongs to this class, except to
say that the doctrine of the lekton is assigned to the topic of ‘that which is signi¬
fied’. 38 He xr]ev yos<; ov xo axexcr) av a<;<;ouvx ocp xrje apiouc xt[>Kec ocp Ae/cra (AA
36 For a more extensive explication of this problem, see Kerferd’s article in Rist 1978: “What
Does the Wise Man Know?” [Kerferd, 1978b). See also the article by Nicholas White, “The
Role of Physics in Stoic Ethics,” Southern Journal of Philosophy: Recovering the Stoics , Spindel
Conference: 1984 (1985): 57-74. White takes the view that “we are not in a position to be
sure why the early Stoics thought that detailed physical and cosmological theory ... would be
required by their ethics” [White, 1985, p. 72). Although White grants the plausibility of the
premiss that they might have held such a view, he argues that any actual arguments for it, or
explanations of it, are lacking [White, 1985, p. 72).
37 At the end of his summary of Stoic logic Diogenes has this to say: “The reason why the
Stoics adopt these views in logic is to give the strongest possible confirmation to their claim that
the wise man is always a dialectician. For all things are observed through study conducted in
discourses, whether they belong to the domain of physics or equally that of ethics (DL 7.83).
Compare: ”[T]he Stoics, who define dialectic as the science of speaking well, taking speaking
well to consist in saying what is true and what is fitting, and regarding this as a distinguishing
characteristic of the philosopher, use [the term ‘dialectic’] of philosophy at its highest. For this
reason, only the wise man is a dialectician in their view" (Alexander in top. ,1.8-14).
38 Contrary to some interpretations (e.g., Kerferd [1978a, p. 260]; Watson [1966, pp. 47-48], we
take it that although every lekton belongs to the class of semainomena, not every semainomenon
is a lekton. Proper names and common nouns, for example, are semainonta which signify ‘in¬
dividual qualities’ (tStot itoioxric) and ‘common qualities’ (xotvfj koiotth;) respectively (DL 7.58).
Since qualities for the Stoics are corporeal, and since lekta are incorporeal, it is evident that both
The Megarians and the Stoics
423
7.63). fie nuAA r]ae pop£ to ctckJj iv xr)E asyusA cov^epvivy xrjE xqeopcjj ocp xr)£ keKzov
av§ xr]£ [iEavivy ocp ttje repp 'to Aextov’, but for the present it will be sufficient to
understood the meaning as ‘what is said’ or ‘what can be said’ (cf. Long [1971],
77); in any case, what is of immediate interest is the reported Stoic classification
of lekta , and in particular, the type of lekton called the axioma.
Axiomata have two characteristic properties which differentiate them from the
other lekta: first, they are the significata of declarative sentences, and second, they
are the only lekta which can be true or false (DL 7.68; cf. AM 8.74). Thus, it seems
apparent that axiomata are somewhat in accord with what we call propositions
(I have already used that term to refer to them in the discussion above); there
are, however, several characteristics with respect to which axiomata differ from
propositions, so that they cannot be merely identified with them (cf. Kneale
and Kneale [1962a], 153-57). It may be, as Long and Sedley propose (1.205),
that ‘proposition’ is the least misleading of the possible translations for axioma ;
nevertheless, we propose to avoid using ‘proposition’ and to merely transliterate
the term.
Having introduced the notion of an axioma, Diogenes goes on to report that
several Stoics, including Chrysippus, divided axiomata into the simple and the
non-simple. Simple axiomata , on this account, are those consisting of one axioma
not repeated (for example: ‘It is day’), whereas non-simple are those consisting
either of one axioma repeated (for example: ‘If it is day, it is day’) or of more than
one axioma (for example: ‘If it is day, it is light’) (DL 7.68-69). Of the non-simple
axiomata, the first introduced is the ‘conditional’, an axioma constructed by means
of the connective ‘if’ (el) (DL 7.71). The Greek word is ‘auvrjpp^vov’, which might
be better rendered ‘connexive’ in accordance with its etymology; however, even
leaving etymological questions aside, translating synemmenon as ‘conditional’ is
somewhat misleading. It seems evident that the Stoic use of the connective ‘ei’
was technical, and hence there are some uses of this connective in ordinary Greek
which seem not to be captured by the Stoic understanding of the term.
One problem with taking ‘conditional’ as the translation of ‘ synemmenon ’ is
that there is a temptation to suppose, as the Kneales seem to do, that what
the Stoics had in mind was to give an account of the occurrence in language
of the connective ‘d’ which would be “satisfactory as a general account of all
conditional statements” (Kneale and Kneale [1962a, p. 135]). It seems to us that
this interpretation gets the matter wrong. The term ''synemmenon'’ denotes a
complex axioma, and according to the description of Diogenes Laertius at 7.66,
this is just to say that it denotes a complex state of affairs; moreover, this complex
state of affairs is signified by the predication of the relation of following between
the states of affairs which are the constituents of the complex axioma. What the
Stoics had in mind was to give an account of the inferences that could be made
given the knowledge that some particular type of event or state of affairs followed
from some other particular event or state of affairs. It seems plausible that they
chose ‘d’ as the syntactic representation of the relation of following because it is
individual qualities and common qualities are semainomena which are not lekta.
424
Robert R. O’Toole and Raymond E. Jennings
suggestive of that relation, as the arrow is suggestive of the relation of implication
in modern syntactic accounts. Hence they chose expressions of the form ‘If A,
B’ to be the canonical representation in their patterns of inference. They might,
however, have chosen to express this relation by saying ‘B follows from A’ rather
than ‘If A, B’, for although they would not recognise the schema ‘B follows from
A; but A; therefore B’ as a proper syllogism of their logical system, they were
nevertheless willing to view it as being equivalent to the syllogistic schema ‘If A,
B; but A; therefore B’, which was an authentic syllogism of their system (Alexander
in an. pr. 373.29-35). And if they had chosen so to represent it, then no one,
we assume, would be tempted to view their characterisation of ‘tot auvrjppevovTa’
as an attempt to give a general account of conditional statements. Having said
all this, we will nevertheless carry on the tradition of translating ‘ synemmenon’
as ‘conditional axidma’, just so long as it is understood that by so doing we do
not assume that in giving a characterisation of ‘synemmenon’ the Stoics supposed
themselves to be providing a general account of conditionals.
Now although the Stoics did not use expressions of the form ‘B follows from
A’ as the canonical expression of the relation denoted by ‘ synemmenon this
notion of ‘following’ in a conditional was of primary importance in their theory
of inference, for their fundamental criterion of a valid argument was based on
this concept. This canon is the so-called conditionalisation principle (cf. Mates
[1953, pp. 74-77]). As it is framed by the Stoics, this principle states that an
argument is conclusive 39 whenever its corresponding conditional is sound (uyie^:
PH 2.137) or true (aXr)AM 8.417): 40 that is, the conditional which has the
conjunction of the premisses as antecedent and the conclusion of the argument
as consequent. Now Sextus Empiricus writes that “the ‘dialecticians’ 41 all agree
that a conditional is sound whenever its consequent ‘follows’ its antecedent” (AM
8.112; cf DL 7.71). In effect, then, one can say that for the Stoics, an argument
is valid (conclusive) just in case its conclusion ‘follows’ from its premisses, as the
consequent follows from the antecedent in a sound or true conditional. In noting
this criterion, however, Sextus also outlines a difficulty, for it seems that although
the ‘dialecticians’ were agreed on the standard for a true conditional, there was
a controversy as to how the notion of ‘following’ was to be characterised (AM
8.112; PH 2.110). There were apparently four competing views, 42 only two of
which, we shall suggest, would have provided a criterion consistent with the role
of inference in Stoic philosophy: the first of these, advocated by those who spoke
39 In some places (e.g., PH 2.137, 146) Sextus uses ‘ouvaxTixoi;’ and ‘douvaxxo!;’ for ‘conclusive’
and ‘inconclusive’ (or ‘valid’ and ‘invalid’), whereas at other places (e.g., AM 8.429) he uses ‘nep-
avTixo?’ and ‘ctitEpavToi;’. Hence, as Mates indicates in his glossary ([Mates, 1953], 132-36), these
terms appear to be interchangeable. Diogenes Laertius, however, in his discussion of arguments
from 7.77-79 uses ‘7tEpavTLxo5’ exclusively for ‘conclusive’ and ‘dxEpavTo?’ for ‘inconclusive’.
40 See page 477 for a discussion of the use of ‘uyie?’ and ‘aXr|c;Ei;’ in these contexts.
41 It is unlikely that by ‘dialectician’ here Sextus is referring exclusively to a member of the
Dialectical School of which Diodorus Cronus and Philo were members. He is probably using it
as a synonym for ‘logician’ (see footnote 20, page 408).
42 See Section 8.1 for a discussion of the four views.
The Megarians and the Stoics
425
of ‘connexion’ (auvdpxrjaic;), required that the contradictory of the consequent
‘conflict’ (paxh™ 1 ) with the antecedent; while the second, advocated by those
who spoke of ‘implication’ (epcpotaic), required that the consequent be ‘potentially
contained’ (TiepLexe™ 1 Suvapei) in the antecedent (PH 2.111-12). 43
The Greek terms used in these contexts are ‘axoXouDeTv’ or ‘eitEicrOoti.’, either of
which has the sense of ‘to follow upon’ or ‘to be consequent upon’. Now although
‘to follow logically’ is no doubt one way in which these terms were understood, it
also seems evident that this meaning is not the only one they carried. But even
if it had been, that fact would not warrant the assumption that for the Stoics
‘to follow logically’ meant quite what it does in a modern setting. For, given
the hypothesis outlined earlier in this section of a motivation for logic grounded in
ethics, we take it that the role of logic in Stoic philosophy is primarily to determine
the inferential relations between states of affairs and only derivatively (if at all)
between sentences of the language. The view we put forward has much in common
with that expressed by A.A. Long in the following passage ([Long, 1971], 95):
The human power of drawing inferences from empirical data presup¬
poses an ennoia akolouthias , an idea of succession or consequence.
... And endiathetos logos, internal speech (reason), is described as
‘that by which we recognise consequences and contradictions’ (x& dxoXoufloc
xod to. paxopeva) But akolouthia is not confined to what we would call
‘logical consequence’. The sequence of cause and effect is explained by
reference to it, for fated events occur xaxa xdyiv xai axoXou^lav[according
to order and consequence] or xaxa xrjv xw v cdxlcnv axoXou<;Lav[according
to the following of causes]. This use of a common term is exactly what
we should expect in view of Chrysippus’ methods of inference from
actual states of affairs.
Given that the world operates according to a strict causal nexus one
of the roles of logic, perhaps its major role in Stoicism, is to make pos¬
sible predictions about the future by drawing out consequences from
the present. The cardinal assumption of the Stoics is that man can
put himself in touch with the rational course of events and effect a cor¬
respondence between them and his own actions and intentions. This
assumption provides the ethical aim of living homologoumenos [harmo¬
niously]. More particularly, ethics is connected with logic and physics
by akolouthia and its related words.
If this is an accurate characterisation of the role of logic in Stoic philosophy, then
in order that such predictions might be carried out, the Stoic conditional will need
to represent the logical as well as the nomic connections, not only between actual
events or states of affairs, but also between mooted events or states of affairs.
Such connections will need to be manifest in the relation between the content of
the antecedent and that of the consequent.
43 Long and Sedley have commented that the containment view “may not differ significantly”
from the conflict view (1.211). We intend to explore this possibility.
426
Robert R. O’Toole and Raymond E. Jennings
In describing these constraints on akolouthia in the above quotation, Long refers
to Chrysippus’ “methods of inference from actual states of affairs” (1971, 95). In
a similar vein, J.B. Gould states that “as Chrysippus maintains, one may gen¬
eralise and affirm that if events of a particular sort occur, then other events of
a specified sort will occur. Such generalisations may be expressed in conditional
propositions. ... These kinds of generalisations, then, are true only when they
denote connections between things or events in nature” ([Gould, 1970], 200-201).
Along the same lines, Michael Frede asserts that “the Stoics seem to regard con¬
sequence and (possibly various kinds of) incompatibility as the relations between
states of affairs or facts in terms of which one can explain that something follows
from something” ([Frede, 1987d], 104). We propose to take seriously this talk of
‘states of affairs’ and ‘facts’ in order to present an interpretation of the notion of
‘following’ or ‘consequence’ in the Stoic system of inference.
If the conception of consequence or ‘following from’ as expressed in the con¬
ditional is a notion of a relation between states of affairs, then to say that a
sound conditional is such that the contradictory of the consequent ‘conflicts with’
the antecedent, as in the ‘connexion’ theory, or that the consequent is ‘potentially
contained’ in the antecedent, as in the ‘containment’ theory, would seem to suggest
that these conceptions of conflict and of potential containment are also notions of
a relation between states of affairs. How is this to be understood? In particular,
how are we to understand this talk of ‘states of affairs’ and the notion of one
state of affairs being ‘in conflict with’ or being ‘potentially contained in’ another?
Unfortunately, there is so little information in the texts about the containment
criterion that we cannot attempt to give more than a speculative account of this
definition. On the other hand, there are fairly clear indications in the texts about
what was meant by ‘conflict’. As to the first part of this question, answering it
will be the burden of our interpretation of Stoic semantic theory. Interpreting
Stoic semantics is in effect to provide an understanding of the theory of the lek-
ton. Moreover, since there is a clear indication in the texts of the dependence of
lekta on ‘rational presentations’ (<pavxaaia Xoyixf]) (e.g. AM 8.70: DL 7.63), it
would seem that an understanding of the lekton will require an interpretation of
this relation in particular, and of the theory of presentations in general; indeed,
it seems to us that there is evidence enough to indicate that one cannot give an
adequate account of Stoic logical theory without taking into consideration the
theory of phantasiai. Such reflections will no doubt entail some involvement in
epistemological questions, and though some writers have ruled out concern with
such questions as being ‘extra-logical’ (e.g., Mates [1953, pp. 35-36]; Kneale and
Kneale [1962a, p. 150]; Mueller [1978, p. 22]), others see them as an essential
component of an understanding of Stoic logical theory (e.g., Imbert [1978, p. 185];
Gould [1970, pp. 49-50]; Kahn [1969, p. 159]).
Our suggestion is that the ‘states of affairs’ which stand in the relation of conflict
or containment in a sound conditional can be thought of as abstract semantic
structures the constituents of which correspond to individuals, properties, and
relations. They are the objective content of rational presentations as well as of
The Megarians and the Stoics
427
axiomata and other types of lekta, and as such are the designata of the Stoic term
1 pragmata’.
4 SEMANTICS
4-1 Epistemology: phantasiai and lekta
The connection between the theory of the lekton and the Stoic doctrine of pre¬
sentations is well documented in the texts. Parallel passages in Sextus Empiricus
and in Diogenes Laertius explicitly refer to this connection. The passage written
by Sextus at AM 8.70 is somewhat more complete:
[The Stoics] say that the lekton is that subsisting (ucptcrrapevov) co-
ordinately with a rational (Xoyixf)) presentation, and that a rational
presentation is one in which it is possible that what is presented be
exhibited by means of discourse. 44
The corresponding passage in Diogenes Laertius is at 7.63: “They (sc. the Stoics)
assert that the lekton is that subsisting coordinately with a rational presenta¬
tion.” 45 The Stoics used forms of the verb ‘ucpioTotaffca’ to indicate a ‘mode of
being’ which is something less than the being that material bodies possess (see
page 462); hence, the use of ‘utpiaiaaffoa’ to describe the being of the lekton , makes
it seems evident that one can take the lekton as somehow dependent for its being
on the corresponding rational presentation. On this understanding of the relation¬
ship between the lekton and the presentation, it is evident that an exploration of
the semantic role of the lekton cannot leave the doctrine of presentations out of
account. As we understand the theory, that which is presented (to cpotviacrdev)
in a rational presentation, and which is capable of being exhibited by means of
discourse, is a pragma (jipaypa). Moreover, we take it that the term 1 pragma ’
is used in Stoic semantics to mean a ‘state of affairs’ which is the unarticulated
objective content of a rational presentation. And since we also understand the
axidma to be what is said in the assertion of a pragma , it would seem necessary
to refer to the Stoic theory of presentations in order to present a characterization
of the notion of akolouthia as a relation between axiomata.
From the point of view of some modern logicians and historians of logic, this
proposal presents a difficulty inasmuch as it implies a merging of epistemological
(and psychological) concerns with logical concerns. Benson Mates, for example,
states in his Stoic Logic that “the criterion for determining the truth of presen¬
tations ... is an epistemological problem and not within the scope of this work”
[Mates, 1953, pp. 35-36], According to the Kneales, “the theory of presenta¬
tions belongs to the epistemology rather than the logic of the Stoics” [Kneale and
Kneale, 1962a, p. 150]. And Mueller expresses the view that considerations of the
44 X6xtov 6e unctpOeiv tpaai to xaxa Xoytxr]v cpavxaaiav txptaxdptevov, Xoyi.xr)v 5k el vat pavxaotav
xa<;’ rjv to (pavxaa^ev eaxi Xoyco Kapaaxr) aai.
45 (pact 8e [to] Xextov sTvai to xaxa (pavxaoiav Xoytxrjv ucptaxa^ievov.
428
Robert R. O’Toole and Raymond E. Jennings
possibility of knowledge of “necessary connections between propositions” (which,
we take it, are a component of the theory of presentations) “would take us outside
the domain of logic and into epistemology” [Mueller, 1978, p. 22], In other words,
there seems to be a strong bias among some contemporary commentators in favour
of the supposition that epistemological considerations could have had no part in
the logic of the Stoa, and that those who cultivated Stoic logic shared this aver¬
sion in common with those who developed the propositional calculus. But not all
present-day commentators agree with this assessment. Some believe, in the words
of Claude Imbert, that “the ancients... judged matters differently” [Imbert, 1980,
p. 185].
Josiah B. Gould, for example, cites textual evidence from Sextus Empiricus
and from Epictetus to support the claim that the Stoics placed their epistemolog¬
ical theory “squarely within the confines of logic” ([Gould, 1970], 49-50). In his
treatise against the logical doctrines of ‘the dogmatists’, Sextus tells us that the
logical branch of Stoic philosophy includes the theory of criteria and proofs (AM
7.24). 46 According to this theory, things which are evident can be apprehended
directly, either through the senses or through the intellect, in accordance with a
criterion of truth. 47 Things non-evident, on the other hand, can be apprehended
only indirectly through the means of signs and proofs by inference from what is
evident (DL 7.25). The inclusion of such a theory—which, as Gould points out
[1970, p. 49], is that of Chrysippus himself—in the logical division of their phi¬
losophy seems clearly to commit the Stoics to consideration of epistemological
concerns within the purview of their logic. As for Epictetus, according to him the
philosophers of the Old Stoa held that logic “has the power to discriminate and
examine everything else, and, as one might say, to measure and weigh them” (disc.
1.17.10). Thus it is “the standard of judgement for all other things, whereby they
come to be known thoroughly” (disc. 1.17.8). This power, it seems to him, is the
reason why these philosophers put logic first in the development of their doctrine
(cf. AM 7.22; DL 7.40). In Gould’s estimation [Gould, 1970, p. 49], these com¬
ments of Epictetus seem to provide further evidence of a close relationship between
logic and knowledge in early Stoic philosophy.
As the quotation above would appear to indicate, Claude Imbert is another
recent commentator who thinks we have evidence that the Stoics took epistemo¬
logical questions to be within the province of logic. She cites the passage at 7.49
in Diogenes Laertius to support this thesis. In an earlier passage Diogenes has
presented a summary of the Stoic logical doctrine (7.39-48), and he proposes now
to give in detail what has already been covered in this introductory treatise. He
begins with a quotation from the book Synopsis of the Philosophers by Diodes of
Magnesia, the passage cited by Imbert:
46 6 8e ye Xoyixoc xoxoc xrjv xepl x£5v xpixrjpuov xoil xGv djioSei^EQv Oewplav TixpisTxev.
47 According to Rist, “the overwhelming body of evidence that we shall consider [concerning
the Stoic criterion of truth] suggests that the normal Stoic answers to the question What is the
criterion of truth? are either Recognition [xaxdXTj<jxc], or Recognizable Presentation [xaxaXr]7ixixf)
(pavxacua]” [Rist, 1969b, p. 133],
The Megarians and the Stoics
429
The Stoics accept the doctrine that the account of presentation and
sensation (aicrOqau;) be ranked as prior [in their logical theory], both
inasmuch as the criterion by which the truth of states of affairs (jipaypaxa)
is determined is of the genus presentation, and inasmuch as the account
of assent (CTuyxaxonlEau;), apprehension (xaxdXqtjjic;), and the process of
thought (vorjau;), although preceding the rest [of their logical theory],
cannot be framed apart from presentation. For presentation comes
first, then thought, being capable of speaking out, discloses by means
of discourse that which is experienced through the presentation (DL
7.49).
This passage, according to Imbert, indicates that “however obscure it may seem to
modern logicians, it is undeniable that the Stoics derived their methods of inference
from certain presentational structures" [imbert, 1980, p. 185]. Moreover, since it
indicates that the criterion of truth is itself a presentation, it also implies that
Stoic logical theory contains an epistemological component.
The difference of opinion among these scholars concerning the content of Stoic
logic is no doubt a reflection of a more fundamental disagreement about its general
nature. Some writers seem to view the Stoic system as an attempt to develop a
calculus of propositions with a truth-functional semantics in accordance with the
model of the propositional calculus which emerged in the twentieth century. For
such writers the inclusion of epistemological (or psychological) components in a
logical system would no doubt be seen as a flaw, a reason to discount such a system
as a genuine logic and to view the attempt at its development as misguided. And of
course, if Stoic logic really were an attempt to develop such a calculus, they would
be right. Other writers, however, seem to proceed with no such presuppositions
about the nature of the Stoic system, or at least, with different ones. This latter
approach is well summarised by C.H. Kahn in a passage which we cited in the
introductory section (see page 400). According to Khan, our picture of Stoic logic
will be distorted if we see it merely as a precursor to the propositional calculus.
A more adequate view would require that we take into account the relationship
in Stoic philosophy between ‘dialectic’ (logic) and their epistemology, semantics,
ethical psychology, and general theory of nature [Kahn, 1969, p. 159]. Since
we are in agreement with this assessment of what is required to construct an
adequate interpretation of Stoic logic in general, we could perhaps appeal to these
remarks as independent justification for including the doctrine of presentations in
an interpretation of the notion of akolouthia. However, assuming that akolouthia
is a relation between axiomata , it would seem that the connection outlined above
between axiomata, lekta , and rational presentations is sufficient in itself to justify
this inclusion.
4-2 Phantasiai
‘Presentation’ (cpavxaaia), according to Stoic doctrine, is an ‘impression’ (xutimgu;)
on the soul or psyche (cj>uxfj) (AM 7.228). Sextus Empiricus attests that this doc-
430
Robert R. O’Toole and Raymond E. Jennings
trine was put in place by Zeno himself (AM 7.2.30; 36), but that it was interpreted
somewhat differently by Kleanthes and Chrysippus (AM 7. 2.28-31). Kleanthes
apparently took the meaning of the term ‘impression’ quite literally, understanding
it in the sense that a signet-ring makes an impression in wax (AM 7.228). Chrysip¬
pus objected to this interpretation, arguing that not only would this model make
simultaneous impressions impossible, but also it would imply that more recent im¬
pressions would obliterate those already in place. Since experience would seem to
show that various impressions can occur simultaneously, and that prior impressions
can coexist with more recent ones, this model cannot be correct (AM 7.228-30).
The model to which Chrysippus appealed was that of the air in a room, which,
when many people speak at once, receives many different impacts and undergoes
many alterations (AM 7.231). The model is apt, as we shall see, since the soul,
according to Chrysippus, is composed of pneuma or ‘natural breath’. Accordingly,
he defined phantasiai as ‘alterations’ or ‘modifications’ occurring in the psyche,
revealing both themselves and that which has caused them: 48 more specifically,
they are modifications of the heqemonikon (nycpovixov), the ‘governing part’ of
the psyche (AM 7.233). 49
We are informed by several sources that the psyche itself has eight parts. 50
Aside from the hegemonikon already mentioned, it consists in the five senses, the
faculty of speech (to <pa>vr)iix6v), and the generative or procreative faculty. 51 In the
account of Diogenes Laertius, the term ‘fiyepovixov’ does not appear; instead, he
uses the term ‘Stavorfuxov’, i.e., the intellectual faculty, “which is the mind itself’
(DL 7.110). 52 The suggestion implicit in this passage is that we can understand
the hegemonikon to be the mind itself; moreover, this interpretation is verified by
Sextus Empiricus in the passage at AM 7.232: “[Presentations occur] only in the
mind or governing part of the psyche.” 53 Thus it would seem that presentations,
according to the Stoics, are modifications or alterations of (or in) the mind. In the
parlance of present-day philosophy of mind, we (at least some of us) would refer
to presentations as ‘mental states’ and associate them with corresponding ‘brain
states’. But for the Stoics there was no need to postulate this kind of dualism (cf.
Sandbach [1971b, p. 10]). For according to them, the psyche is constituted by
1 pneuma' or ‘breath’ (uveupa), itself composed of the elements fire and air “which
are blended with one another through and through” (Galen SVF 2.841). 54 It seems
evident on this account that since fire and air are material elements par excellence,
the soul must also be a material entity.
48 Sextus Empiricus AM 7.230; Aetius SVF 2.54. Sextus uses the term ‘rjtEpouiaEi.?’, which we
have rendered as ‘alteration’ or ‘modification’ (cf. Bury). In the corresponding passage, Aetius
uses ‘gpa’joc’, which might be rendered ‘affection’ (cf. Long and Sedley [1990, 1.237]).
49 This expanded definition was put forward in order to forestall certain objections that not all
modifications of the psyche could be presentations.
50 Nemesius SVF 1.143; Chalcidius SVF 2.879; DL 7.110.
51 to yevviynxov: DL 7.110; to aitepparuxov: Nemesius SVF 2.39.22.
52 OXEp ECTTIV aUTT| f) SlOtVOlCC
53 aXXa tie pi xfj Siavoia povov xai t£5 f)YEpovixa>.
54 8d oXcov aXXrjXoic xexpappeva.In his Alexander of Aphrodisias on Stoic Physics , Robert B.
Todd discusses the Stoic theory of total blending as it is reported by Alexander in De mixtione.
The Megarians and the Stoics
431
The Stoics held that there are two principles (otpxou) i n the universe: the passive
(to Kotaxov), which is substance without quality (oucua rcoia), or prime matter
(uXr)), and the active (to Ttoiouv), which is the logos inherent in matter, or God
(DL 7.134). By the nature of the properties ascribed to the pneuma, it would
appear that the active principle is embodied in it. First, the pneuma is the force
which maintains the universe as a unity. Chrysippus, for example, holds that “the
whole of substance is unified because it is entirely pervaded by a pneuma, by means
of which the universe is held together, is maintained, and is in sympathy with
itself.” 55 Second, the pneuma invests with qualities the undifferentiated matter
(uXr)) in which it inheres (Plutarch de Stoic repugn. 1054a-b). And third, the
pneuma is constitutive of the souls of human beings. According to Chalcidius,
Zeno and Chrysippus put forward similar arguments for this thesis. Chrysippus
argues thus: “It is certain that we breathe and live with one and the same thing.
But we breath with natural breath ( naturalis spiritus). Therefore we live as well
with the same breath. But we live with the soul. Therefore, the soul is found to
be natural breath” (SVF 2.879). 56
According to the Stoics, then, the psyche is corporeal, and hence just as much a
material entity as is the substantial body of which it is a part. Further arguments
for this thesis are set out both by Kleanthes (SVF 1.518) and by Chrysippus
(SVF 2.790); moreover, there is no doubt that they followed Zeno in this view
(SVF 1.137; 138; 141). The details neither of these arguments nor of the under¬
lying physical theory need concern us here; what is of interest, however, is the
implication that when the Stoics speak of modifications or changes in the mind,
they are not speaking metaphorically. A modification of the mind would appear
to be a determinate change of state of the mind-substance or pneuma (nvEupa).
Hence a presentation, since it is such an alteration, would be a physical event (cf.
[Sandbach, 1971a, p. 10]), as much a physical entity as the pneuma itself. 57
At 7.51, Diogenes Laertius provides evidence that the Stoics observed a dis¬
tinction among presentations between those which are sensory (aia’drjii.xai) and
those which are non-sensory (oux alcTdrjTixaf). “Sensory impressions,” according
to Diogenes’ account, “are those which are apprehended (Xappavopevai) through
one or more of the sense organs; non-sensory, on the other hand, are those which
apprehended through thought or by the mind (Sta Tfjc Siavotac), such as those of
the incorporeals (daupaxa) and other things apprehended by reason (DL 7.51).
We take it that this distinction is designed to describe presentations in accordance
with the character of their immediate sources. Obviously, a sensory presentation
55 eaxi 8e r) XpuaiKxou 8ol;cx Kepi xpaaeoc fj8e rjvOa'Sai pev OxoxMlexcxi xrjv aupxaaaav ouaiav,
Ttveupaxoc xivoc; 8ia naar]C auxfjc Sirjxovxoc, ucp’ ou csuveyExcu te xai auppivei xai aupKaOec eaxiv
aOxfi xo 7iav(Alexander de mixtione 216.14-17).
56 Zeno’s corresponding argument is also recorded by Chalcidius (SVF 1.138). cf. Tertullian
(SVF 1.137).
57 0ne fragment seems to indicate that not only presentation, but also assent (auyxaxadeaic),
impulse (oppr)), and reason (Xoyoi;) are qualities of the psych (Iamblichus de anima, apud
Stobaeus ecloq. 1.368.12-20; cf. AM 7.237), and qualities, according to the Stoics, are corporeal
(SVF 2.376-98).
432
Robert R. O’Toole and Raymond E. Jennings
is one whose immediate source is an actual state of affairs, and this state of affairs
is also its cause. It is somewhat unclear what the immediate source of non-sensory
presentations might be, but since they are “apprehended through thought,” per¬
haps the most likely candidate would be another presentation. Moreover, there
would seem to be nothing to stand in the way of this second presentation’s being
the cause of the first, for presentations, as was noted above, are ‘bodies’ (somata)
and hence can enter into causal relationships (Stobaeus eclog. 138.24; AM 9.211).
Indeed, there would seem to be no reason why one could not envisage a sequence
of presentations forming a causal chain.
Ultimately, however, there must be a presentation whose cause is not another
presentation, but rather some external state of affairs. This requirement would
not be a problem in the case of some non-sensory presentations: for example, the
Stoics hold that “it is not by sense-perception (atcn9r|ai<;) but by reason (Xoyoe;)
that we become cognizant of the conclusions of demonstrations, such as of the ex¬
istence of the gods and of their providence” (DL 7.52). Presumably, since the gods
themselves are evidently corporeal entities, 58 it would be as a result of their actions
that one would become cognizant of their existence, and the presentation in which
one apprehends that existence would have its cause in the gods themselves. But
in the case of other non-sensory presentations, such as those of the incorporeals,
there is a difficulty in seeing how to give an account of the causal basis of such
a presentation. For the Stoics hold that the class of incorporeals, which includes
lekta, void, place, and time (AM 10.218), are asomata (literally ‘without body’)
and hence cannot enter into causal relationships (AM 8.263). A plausible solution
to this difficulty is suggested by Long and Sedley, who propose that “perhaps we
should connect [this relation between asomata and presentations] with ‘transition’
[jicidpaCTic], a method by which incorporeals are said to be conceived” ([Long and
Sedley, 1990], 1.241). They go on to suggest that “this refers ... to the mind’s ca¬
pacity to abstract, e.g., the idea of place from particular bodies” [Long and Sedley,
1990, 1.241]. In an earlier work, however, Long renders metabasis as “a capacity
to frame inferences” [Long, 1971, p. 88], and there are several texts which would
seem to confirm this reading. 09 Presumably, such a capacity would be seated in
the mind ( hegemonikon ) or intellectual faculty ( dianoetikon ) and thus ultimately
in the soul itself (cf. DL 7.110).
According to Iamblichus as quoted by Stobaeus, “those philosophers who follow
Chrysippus and Zeno and all those who conceive of the soul as body, bring together
58 Aristocles SVF 1.98; Chalcidius 293, L & S 44E; Galen hist, phil., DDG 608; DL 7.134.
One reading of DL 7.134 (Suidas) has it that the archai are aacopaxa(incorporeal). The reading
of the codices, however, has it that they are atopaxa(corporeal). Long and Sedley prefer the
reading ‘atopaxa’ [1990, 2.226], their reasons being (1) that this interpretation is supported by
other texts, and (2) the corporeality of the principles follows by implication from the Stoic view
that only bodies are capable of acting and being acted upon [Long and Sedley, 1990, 1.273-74],
But since the active principle is explicitly identified with God (DL 7.134), then it would seem to
follow that God (or the gods) is corporeal.
59 AM 8.194;275;3.25;DL 7.53. Sandbach [1971a, p. 26] translates 1 metabasis’ in DL 7.53 as
‘inference’.
The Megarians and the Stoics
433
the powers (od Suvctpeic) of the soul as qualities in the substrate (uTtoxetpevov),
and posit the soul as substance (ouo(a) already underlying the powers” ( eclog.
1.367.17). Moreover, there are several texts which report that the Stoics charac¬
terise qualities as corporeal, 60 and at least one passage specifically reports that
they describe the qualities of the psyche as such (Alexander de amnia 115.37).
Given their corporeal nature, one might suppose that the various capacities of
the psyche would have causal powers, and this supposition gains credence from a
passage in which Zeno is reported to hold that prudence (cppovqmc) is the cause
of ‘being prudent’ (to cppovetv), and temperance (owcppoauvr)) is the cause of ‘be¬
ing temperate’ (to awcppovelv) (Stobaeus eclog. 1.138). Thus one might plausibly
conjecture that the Stoics could give an account of the causal basis of non-sensory
presentations, such as those of the incorporeals, by invoking, presumably along
with the data of experience, the causal powers associated with metabasis. And
one might further suppose that some such account would throw light on the Stoic
explanation that “presentations are formed because of [the incorporeals] and not
by them,” 61 and that they are perceived not by the senses, “but in a certain man¬
ner by the senses” (sed quodam modo sensibus) (Cicero acad. 2.21). If so, then we
would not have to suppose with Long and Sedley that these explanations represent
an attempt by the Stoics “to find a relationship other than causal to fit the case”
[Long and Sedley, 1990, 1.241],
Whatever may be the difficulties involved in providing a causal basis for non-
sensory presentations, no comparable problems exist for sensory presentations, for
the source of such presentations is an actual state of affairs. We can probably take
a passage of Aetius to imply that sensory presentations are the primary means by
which a person develops the stock of conceptions which comprise the content of
memory and experience.
When a man is born, the Stoics say, he has the commanding-part of his
soul like a sheet of paper ready for writing upon. On this he inscribes
each one of his conceptions (evvoiai). The first method of inscription is
through the senses. For by perceiving something, e.g., white, they have
a memory of it when it has departed. And when many memories of a
similar kind have occurred, we then say we have experience (epitetpla).
For the plurality of similar impressions is experience. Some conceptions
arise naturally in the aforesaid ways and undesignedly, others through
our own instruction and attention. The latter are called ‘conceptions’
only, the former are called ‘preconceptions’ (TipoX^eu;) as well. Rea¬
son, for which we are called rational, is said to be completed from our
preconceptions during our first seven years. 62
60 For example, Plutarch de comm. not. 1085e; Galen SVF 2.377; 410; Simplicius in cat.
271.20; in phys. 509.9. Long and Sedley argue indexLong, A. A.that “the corporeality of qualities
is one of many Stoic theses implied by the corporeality of the principles” [Long and Sedley, 1990,
1.274] (see 432, footnote 58).
01 ex’ ocutoTc (gavTaaioupcvou xa\ oux On’ auxSv (AM 8.409).
02 Aetius plac. 4.11.1—4, DDG 400 = SVF 2.83. The translation is that of Long and Sedley
434
Robert R. O’Toole and Raymond E. Jennings
Another distinction among presentations which is relevant at this point is that
between presentations which are rational (XoyLXotl) and those which are irrational
(aXoyoi). Rational presentations, according to Diogenes Laertius, are those of
rational creatures. They are processes of thought (DL 7.51), and they have an
objective content which can be expressed in language (AM 8.70). It looks as
though the ‘preconceptions’ mentioned in the quotation from Aetius are those
which ‘arise naturally’ from sensory presentations. Since these preconceptions
are a requisite for rationality, it is apparent that our first sensory presentations
are preconceptual and hence non-rational. Evidently, rational presentations are
possible only when a person has acquired the preconceptions which go to make
up the content of such presentations. Since the preconceptions would seem to
provide a fairly basic conceptual apparatus (e.g., colour concepts), the rational
presentations based on them would also be fairly basic. Diogenes Laertius lists
several ways that more complex conceptions may be brought about.
Of these [complex conceptions] some are acquired by direct experience,
some by resemblance, some by analogy, some by transposition, some by
composition, and some by contrariety. ... Some things are conceived
by inference (pcxapacnc;), such as lekta and place. The conception of
what is just and good comes naturally. And some things are conceived
by privation, such as the idea of being without hands (DL 7.52-53).
Sextus Empiricus gives a similar list of ways by which conceptions are grasped, and
it is notable that he precedes this list with the comment, apparently having its basis
in Stoic doctrine, that “in general it is not possible to find in conception that which
someone possesses not known by him in accordance with direct experience” (AM
8.58). Of the rational presentations which are primary, an important sub-class are
those presentations which are called ‘apprehensive’ or ‘cognitive’ (cd xaxotXr)7txixai
(pavxc(CTiai).We follow F.H. Sandbach in rendering ‘ai xaxaXrjxxixod cpatvxacrfai’ as
‘cognitive presentations’ [Sandbach, 1971a, p. 10]. Presentations belonging to this
class play a central role in the Stoic theory of knowledge.
The material nature of the mind in Stoic psychology is an important component
in what seems to us a plausible interpretation of the notion of an ‘apprehensive’ or
‘cognitive’ presentation. The interpretation we have in mind is that presented by
Michael Frede in his essay “Stoics and Skeptics on Clear and Distinct Impressions”
[Frede, 1987e, 151-76]. Cognitive presentations (Frede calls them ‘cognitive im¬
pressions’) were deemed by the Stoics to be “the criterion of truth” (xo xptxrjpiov
[1990, 1.238]. According to Sandbach, “the claim that reason is made up in the first seven years
is surprising and conflicts with all other sources, which give 14 as the age when it is established.
.. . Au)’tius seems to have confused the beginning of the growth of reason in the first seven
years of life with its completion round about the age of fourteen” [Sandbach, 1985, 80nll8].
Jamblichus, for example, reports that “the Stoics say that reason is not immediately implanted,
but is assembled later from sense perceptions and presentations about the fourteenth year” (de
anima, apud Stobaeus eclog. 1.317.20). cf. Inwood: “Reason ... begins to be acquired at or
about the age of seven and is ‘completely acquired’ at or about the age of fourteen” [Inwood,
1985, p. 72],
The Megarians and the Stoics
435
xrjc aXrydElac) (DL 7.54), and as such played a foundational role in the Stoic account
of the development of an individual person’s knowledge of the world. According
to the definition given both by Sextus Empiricus (AM 7.248) and by Diogenes
Laertius (7.46), cognitive presentations arise only from that which is real and are
imaged and impressed in accordance with that reality. 63 In his account, Sextus
Empiricus adds a third condition to this definition: a cognitive presentation can¬
not have its source in that which is not real (AM 7.248). This last condition was
apparently added to forestall certain objections of the Academics. According to
Sextus (AM 7.252), the Stoics thought that a cognitive presentation would pos¬
sess a distinctive feature (ISlupa) by which it could be distinguished from all other
presentations, such a feature reflecting a corresponding distinction in the object
from which the cognitive presentation arises. The Academics, on the other hand,
denied that any presentation could have such a feature. According to them, a
false presentation can always be found which is similar in all respects to any given
presentation (AM 7.402-10). According to Frede, both the Stoic and Academic
schools probably agreed that cognitive presentations, “in order to play the role
assigned to them by the Stoics, would have to satisfy the third condition too”
[Frede, 1987e, pp. 165-66].
A problem for the Stoics, then, is to give an account of how one could tell that
a presentation satisfied this condition, or, as Sandbach expresses it, “How could
the bona fides of a cognitive presentation be established?” [Sandbach, 1971a, p.
19]. The difficulty is that any sort of test one might make to determine whether
a given presentation is cognitive will itself depend on a presentation. But then
a test will be required to determine whether the latter presentation is cognitive,
and so on. Evidently such a process will lead to an infinite regress, a criticism
expressed by Sextus Empiricus (AM 7.429), and probably derived from the early
Academics, Arcesilaus and Carneades (cf. Long and Sedley, [1990, 1.249]).
One sort of reply to this criticism is that proposed by Sandbach: “There must
be a point to call a halt. There must be some presentations that are immediately
acceptable, that are self-evidently true. That is what constitutes a cognitive pre¬
sentation. It is the attitude of common sense that most presentations are of this
sort” [Sandbach, 1971a, p. 19]. It is not clear how far this reply will go to convince
the sceptic. At any rate, if there were to be such self-evidently true presentations,
it seems a plausible supposition that they would be sensory presentations having
fairly basic conceptions as content. From these basic cognitive presentations the
corresponding conceptions would be derived, and from these, in turn, more com¬
plex presentations. Thus, through the development of more and more complex
notions, a complete grasp of things would eventually be gained, such grasp being
expressed in general conditionals such as this: “If a thing is a human being, it is
63 We are rendering ‘to ujiotpxov’ as ‘that which is real’ or ‘reality’ rather than as ‘the real
object’. We will argue in the sequel that although the Stoics took “objective particulars” as
their “fundamental existents” (cf. Long [1971, p. 75]), they nevertheless thought that reality
consists not only in such objective particulars, but also in the properties and relations of these
objects.
436
Robert R. O’Toole and Raymond E. Jennings
a rational mortal animal” (Si homo est, animal est mortale, rationis particeps).
This seems to be the developmental process envisaged by Antiochus in his defence
of Stoic epistemology (Cicero acad. 2.21). Frede conveys the idea with the remark
that “the Stoics take the view that only perceptual impressions are cognitive in
their own right. Thus other impressions can be called cognitive only to the ex¬
tent that they have a cognitive content which depends on the cognitive content of
impressions which are cognitive in their own right” [Frede, 1987e, p. 159],
Frede suggests that these basic presentations which are self-evidently true are
so because they possess a causal feature which acts on the mind “in a distinctive
way” thus bringing about recognition of the veridicality of the presentation [Frede,
1987e, p. 168]. “It is in this sense,” according to Frede, “that the mind can
discriminate cognitive and noncognitive impressions” [Frede, 1987e, p.168]. The
plausibility of this suggestion, it seems to us, depends in no small measure on the
material nature of the mind in Stoic psychology. Previously in this section we saw
that the pneuma which pervades all substance is also constitutive of the minds of
human beings. Now according to the Stoics, causal interactions between bodies
occur either through spatial contact (Simplicius in cat. 302.31) or through the
medium of the pneuma (Aetius plac. 1.11.5, DDG 310). Hence, the feasibility of
a causal interaction between the mind and some distinctive feature of a state of
affairs is not prima facie out of the question; moreover, such an interaction would
evidently result in a unique presentation.
5 LEKTA
5.1 Signifier and Signified
Traditionally, one of the more celebrated texts providing evidence for Stoic seman¬
tic theory is that presented by Sextus Empiricus at AM 8.11-12. Just before this
passage he has given an account of a controversy between the Epicureans and the
Stoics as to whether the true is that which is perceptible only to the senses or only
to the intellect. He continues:
Such, then, is the character of the first disagreement concerning what
is true. But there was another controversy according to which some
located both the true and the false in that which is signified, some in
the utterance, and some in the process of thought. The Stoics, more¬
over, put forward the first opinion, saying that three things are con¬
nected: that which is signified (to crrjpaivopevov), that which signifies
(to or|[idivov), and the subject of predication (to Tuyydvov). Of these,
that which signifies is the utterance (cpovrj), for example, ‘Dion’. That
which is signified, that is, what is indicated (to SqXoupevov) by the
utterance, is the state of affairs itself (auTO to xpctypa) which we ap¬
prehend as subsisting coordinately with (xapacpuxrapevou) our thought,
but which the Barbarians, although hearing the utterance, do not com¬
prehend. The subject of predication is the external substrate (to extoc
The Megarians and the Stoics
437
UTtoxdjievov) as, for instance, Dion himself. And of these (three things)
they say that two are corporeal, namely, the utterance and the subject
of predication; whereas one is incorporeal and spoken, or able to be spo¬
ken (Xextov), namely, the state of affairs (Ttpaypa) signified, precisely
that which also becomes (yivexat) true or false. And these ( pragmata
which are spoken or can be spoken) are not all of a kind, but some
are incomplete (eXXutfj), while others are complete (auTOxeXrj). And of
the complete, one is called axidma, which they also describe by saying
“The axidma is that which is true or false.”
The above passage provides a point of reference for the discussion of various ques¬
tions which play a central role in the interpretation of Stoic semantic theory. It
is our intention that an understanding of this theory will emerge as a result of
discussing these various issues. Since we will frequently refer to this passage in
what follows, it will be convenient for such reference to call it ‘Passage A’.
Sextus informs us in this passage that the Stoics develop their theory of what
is true or false by distinguishing three kinds of items which are connected. These
are ‘that which signifies’ (to arjpaivov), ‘that which is signified’ (to ar]pai.vopevov),
and ‘the subject of predication’ (to tuyx&vov). He goes on to provide a more
specific delineation of each of kind of item. That which signifies (to semainon ) is
characterised as ‘the utterance’ (f) cpcovfj). The term ‘cpcoviq’ is standardly rendered
as ‘sound’ or ‘speech’, but it seems to us that in certain contexts it has a somewhat
more ambiguous meaning for the Stoics, this meaning being better captured by the
indeterminate sense that ‘utterance’ has as it is used in modern philosophy. For
instance, ‘utterance’ on this account would encompass writing as well as speech (cf.
DL 7.56). As an example of to semainon, Sextus provides the utterance of the name
‘Dion’. This example, it seems to us, is not only completely inappropriate for the
context, but is also inappropriate at a more fundamental level. We shall have more
to say about this problem presently. That which is signified (to semainomenon),
that is, what is indicated by the utterance, is characterised as ‘the state of affairs
itself’ (cxuto to xpaypa). 64 Sextus implies that on hearing the utterance a Greek
speaker will apprehend the pragma as ‘subsisting coordinately with thought’, but
the Barbarian or non-Greek-speaker will not apprehend the pragma , even though
he hears the same utterance. Recalling that rational presentations are ‘processes
of thought’ (DL 7.51) having an objective content which can be expressed by
discourse (AM 8.70), this sounds very much like a description of how a rational
presentation would be induced in the mind of the Greek speaker by the utterance,
with the content of the presentation being the state of affairs signified by the
utterance.
64 We follow several authors in translating ‘to nparpoi’ as ‘the state of affairs: e.g., Long [1971,
107nl0]; Long and Sedley [1990, 1.195, 202]; Reesor [1989, Ch. 3]. The term certainly can have
this meaning in ordinary Greek; it seems evident, however, that the Stoics gave it a technical
meaning in the context of their semantic theory. We take it that in this context the term referred
to a semantic structure which corresponded in structure either to a real state of affairs, or to a
mooted state of affairs.
438
Robert R. O’Toole and Raymond E. Jennings
To return to the problem of Sextus’ example ‘Dion’, the context of Passage A
is an account of what it is to which the Stoics ascribed the property of having a
truth value, and we are told that it is to semainomenon. We are also told that to
semainomenon is the pragma or state of affairs indicated by the utterance. Now
it seems evident that the utterance of ‘Dion’ will not indicate a state of affairs
which is either true or false. Hence, the example seems to be inappropriate in the
context of the discussion. A more suitable example would be something like the
utterance of the sentence ‘Afiov xEptxaxet’ (Dion is walking). 65
At a more fundamental level, the example is problematic inasmuch as it seems to
suggest that in Stoic semantics a proper name signifies a ‘meaning’ or ‘sense’ and
refers to the object named. According to Diogenes Laertius, however, Diogenes
the Babylonian defined a name as a part of speech (pepo<; Xoyou) indicating an
individuating quality (tSia 7toioTr)<;) (DL 7.58). This teaching would appear to have
its basis in certain epistemological and metaphysical concerns, in particular, in the
doctrine of cognitive presentations, in the principle of the identity of indiscernibles,
and in the theory of change and identity. Recall that one of the functions of the
pneuma in Stoic philosophy is to invest undifferentiated matter (OXr)) with qualities
(see page 430). Certain of these qualities serve not only to differentiate portions
of prime matter from the rest, but also to individuate them as unique entities.
The matter invested with an ‘individuating quality’ (LSiot tcoiott)?), along with
the quality itself, together comprise ‘that which is individually qualified’ (6 I8ia>c
itoiov), that is, the uniquely qualified individual which serves as the substrate for
further qualities and for the predication of attributes (Simplicius in cat. 48.11).
An essential feature of this doctrine is that although the substance (oucna) of which
an individual entity is comprised is constitutive of that entity, it is not identical
with it (Stobaeus eclog. 1.178.21-179.17). Thus, the Stoics were able to defend the
idea of something which remains constant and serves as the basis for change, for
although the substance of which an entity is comprised might undergo constant
‘alteration’ (aXXokoau;) and so never be the same from moment to moment, the
individuating quality remains constant (Stobaeus eclog. 1.177.21-178.21; Plutarch
comm. not. 1083c). As well, this notion of a uniquely qualified individual is no
doubt the basis of the Stoic thesis of the identity of indiscernibles which held that
“no hair or grain of sand is in all respects the same as another hair or grain”
(Cicero acad. 2.85), and which served in the defence of the theory of cognitive
presentations (acad. 2.83-85).
According to this doctrine, then, the utterance of ‘Dion’ signifies the portion
of pneuma individuating that part of the substrate (to UTtoxcipEvov) which is
constitutive of the qualified individual (iSitoc tcoioc), Dion. And even when Dion
has died and it is no longer possible to refer to him by means of a demonstrative,
65 For similar commentary on Sextus’ example, see Long and Sedley [1990, 2.197]; Long [1971,
p. 77 and 107n 11]; Frede [1987b, p. 349]. Kerferd, on the other hand, argues that the conclusion
that ‘Dion’ signifies a lekton is straightforward in spite of the many passages suggesting that
only axiomata are true or false [Kerferd, 1978a, pp. 260-61], He does not mention the difficulty
posed by Diogenes’ passage which says that names signify corporeal qualities, not incorporeal
lekta.
The Megarians and the Stoics
439
it is still possible to refer to him by name (Alexander in an. pr. 177.31), since the
name picks out not the substance of Dion, but the individuating quality. At any
rate, the point is that the Stoics already have an adequate theory of signification
for names which links the utterance of the name directly to what it signifies, and
there is no need, therefore, to posit an incorporeal ‘meaning’ or ‘sense’ as the
signification of a name.
Returning to Sextus’ account of the three connected items, the ‘subject of
predication’ (to TUYxavov) is characterised as ‘the external substrate’ (to extoc
UTcoxd[iEvov). According to Simplicius, the Stoics, as well as earlier philosophers,
held that the substrate is twofold: primarily it is unqualified matter (cbtoioc uXri),
which is what Aristotle named it; and secondly, it is that which is commonly or
individually qualified (6 xoiv«c koiov q IStox;). In this latter case, the qualified
substrate serves as the substrate for further qualities and as the subject of pred¬
ication (in cat. 48.11-16). Since it seems evident that the utterance signifies the
pragma and predicates a property or quality of the external substrate, we have
translated ‘to tuyx«vov’ as ‘the subject of predication’. 66
Having given this more specific characterisation of the three connected items,
Sextus then reports the Stoic doctrine that two of these items are ‘corporeals’
(ooporca), which is to say, bodies or material entities, while the other is ‘incor¬
poreal’ (aoQpa), literally, ‘without body’. It seems obvious that the Stoics would
have classed the external substrate as corporeal. Moreover, since they viewed the
utterance as a body, it seems clear that they would also have classed it as corpo¬
real. It is Stoic doctrine that whatever produces an effect is a body; hence, the
utterance is evidently a body, for it produces an effect as it proceeds from the
speaker to the hearer (DL 7.55-56). In addition, since a written utterance (cpcnvf]
eYYP < W aT °c)—which, according to Diogenes the Babylonian, is speech (X^u;) (DL
7.56)—is also capable of producing an effect, they no doubt would have counted
it as corporeal as well. On the other hand, given that the Stoics held that “bodies
alone are existents” (Plutarch comm. not. 1073e), the conception of the pragma
as incorporeal does seem to be problematic, for, as Gerard Watson puts it, “‘in¬
corporeal’ is an extraordinary concept in a materialist universe” [Watson, 1966,
p. 38], We shall have more to say about the ontological status of the pragma
or lekton in Subsection 5.5. For the moment, however, we discuss the semantic
considerations which might have prompted the Stoics to posit such an item.
At the end of Passage A Sextus makes it clear that there are several different
types of complete pragmata, and in a later passage (AM 8.70-74) he provides a list
of them. In this later text, however, he does not write, as he does in Passage A,
of complete pragmata which are spoken or can be spoken (Xextov), but rather of
complete lekta. It would appear, therefore, that we can take complete lekta to be
complete pragmata which are spoken or can be spoken. From what was said earlier
66 Long and Sedley translate ‘to tuyx“' , ov’ as ‘the name-bearer’. For an explanation and
discussion of this translation, see Long and Sedley [1990], 1.201 and 2.197. in an earlier work,
Long translated ‘to tuyxcxvov’ as ‘the object of reference’ [Long, 1971, p. 76, 107n9]. For Michael
Frede’s interpretation, see his [1987b, pp. 349-50],
440
Robert R. O’Toole and Raymond E. Jennings
in Passage A, the lekton is the signification of an utterance, and from discussion
in the text at AM 8.70-74, as well as in the text of Diogenes Laertius at 7.65-68,
the type of the lekton is evidently determined by the type of speech act which is
its signifier. One type of lekton , for example, is signified by the utterance of a
command, another type, by the utterance of a question (DL 7.66; AM 8.71). The
axioma, as one might expect, is apparently signified by an assertion, which is to
say, the utterance of a declarative sentence (DL 7.65-66).
Now it would seem that a difficulty becomes manifest when one asks what is
the character of the lekton, or, as it might be expressed, what is the character of
‘that which is signified’. This difficulty is relevant to every type of lekton , but one
can get a general idea of the problem by considering the axioma in particular. At
the beginning of Passage A Sextus informs us that the Stoics rejected the view
that the true and the false are in the utterance, as well as the view that they are
in the process of thought. They themselves put forward the thesis that the true
and the false are located in ‘that which is signified’ (to orjpaivojiEvov) . At the end
of Passage A, we are told that of the various types of complete lekta , the axioma
is the one which the Stoics say is true or false, and according to Diogenes Laertius
(7.65-66), the axioma is signified by an assertion. Hence, according to the Stoics,
when someone makes an assertion, i.e., utters a declarative sentence, he signifies
an axioma , and the axioma is either true or false. The problem, then, which in
general can be expressed as ‘What is the nature of “that which is signified”?’, can
be expressed with respect to the axioma as ‘What is the nature of “that which is
true or false”?’.
It might be helpful at this point to consider how the Stoics define something as
being true. Sextus reports in one place, for example, that they hold the definite
axioma ‘This man is sitting’ or ‘This man is walking’ to be true (aXrfdEc) whenever
the predicate ‘to sit’ or ‘to walk’ corresponds to the attribute falling under the
demonstrative (AM 8.100). 67 Similarly, Diogenes Laertius relates that on the Stoic
account, someone who says ‘It is day’ seems to make a claim that it is day, and
the axioma set forth is true (dXryfiet;) just in case it really is day, otherwise, it is
false (c^euSoc) (DL 7.65). 68
Now suppose that someone utters the sentence ‘Dion is walking’, and suppose
further that Dion really is walking. Evidently, on the above account, what is
signified by the utterance would be true, and it seems tempting in such a situation
67 xod 8e to &>pia[xevov touto &5!<ona, to ‘ouxop xtzOiprai’ i) ‘ouxot; nepuiaxeT,’ tote cpaaiv olAt^ec
uncxpxeiv oxav xfi uxo xf)v SeT^iv tu'xtovti au(i[3e[3)jxT) to xaTT)Y6pr)(ia, oTov to xatifjCT'dai i] to xepi-
xocteTv.
68 Note that in English, the axioma is mentioned by setting of the corresponding sentence in
single quotation marks. In the Greek, it is often mentioned by similar means—usually with
double quotation marks, and examples are often introduced by the term ‘oTov’, with or without
quotation marks. Also, the axioma is sometimes mentioned by nominalising the corresponding
sentence by means of the definite article. Hence, the axioma which we represent in English as
‘Dion is walking’, may be represented in the Greek as ‘to A!<jv xEpixaTEf’. At any rate, the same
means are used to mention sentences, both in English and in the Greek, respectively, and it
would appear that some ancient commentators, as well as some recent translators, do not always
keep the distinction in mind.
The Megarians and the Stoics
441
to think that what is signified is the actual state of affairs, which, on the Stoic
view, could be described as Dion’s hegemonikon in a certain state. And since the
hegemonikon, which part of the soul, is constituted by pneuma and so is corporeal
(see page 430), what would be signified on this understanding would be something
corporeal, and hence unproblematic for Stoic materialism. 69 Suppose, however,
that someone utters the sentence ‘Theon is walking’, and that Theon is actually
sitting at the Stoa listening to Zeno’s lecture. Evidently, what is signified by the
utterance in this case will be false; moreover, there is no temptation to think
that what is signified is an actual state of affairs consisting in the non-walking
Theon. Nevertheless, the utterance is significant, and whether Theon is actually
walking or not, a Greek speaker who hears the utterance of the sentence ‘Qewv
TtepuiaxEL’ will experience in either case a rational presentation according to which
he will apprehend the same pragma signified and spoken. In other words, what
will be signified by the utterance will be the same in either case. It seems evident,
therefore, that what is signified is not the actual state of affairs. What is suggested
by this commentary is that the Stoics were persuaded by theoretical considerations
to admit items into the ontology of their theory of language for which they could
not give a materialist account.
It may be, however, that because of reflections on their theory of causality,
the lekton , or at least, the incomplete lekton, had already been admitted as an
item in their ontology. Although the weight of evidence adduced by many modern
commentators would seem to support the view that the lekton was posited by the
Stoics in their semantic theory, Michael Frede has recently proposed that “it is not
clear ... that the notion of a lekton was introduced by the Stoics in the context
of their philosophy of language rather than their ontology” [Frede, 1987a, p. 137].
The evidence for this proposal comes from a passage in Clement’s Stromata (8.9.4)
in which it is claimed that Kleanthes called predicates (xaxrjyoprinaxa) lekta. As far
as we know, according to Frede, this is the first use by a Stoic of the term ‘lekton’
[Frede, 1987a, p. 137]. In order to bring out the significance of this passage with
respect to the present concern, it will be necessary to consider briefly the role of
the predicate (kategorema) in the Stoic theory of causality.
It would seem that for the Stoics “the canonical representation of the causal
relation was ... as a three-place relation between a body and another body and
a predicate true of [the second] body” (Frede [1987a, p. 137]). Thus a knife (or
a scalpel) is the cause for flesh of being cut (Tfj oapxt too xEpvecrdoa) (AM 9.211;
Clement strom. 8.9.30.3), and fire is the cause for wood of burning (x£> £6Xo
xoO xaiecrdoa) (AM 9.211). In representing the causal relation in this manner the
Stoics were no doubt influenced by their conception of the universe as a dynamic
continuum. Such a view would seem to presuppose a theory of causality in which
events rather than particular entities are seen as the effects of causes. For on this
conception, the universe just is the totality of events which occur as the result
69 It is evident that the Stoics thought that the attribute ‘walking’ is real when possessed by
someone or something, even though they also thought that the predicate ‘walking’ is incorporeal
(see footnote 84, page 448).
442
Robert R. O’Toole and Raymond E. Jennings
of causal interactions between bodies (somata), either through spatial contact or
through the medium of the pneuma. 70 According to Sextus Empiricus, the Stoics
characterise such interactions as follows: “Every cause is a body which becomes
responsible to a body of something incorporeal” (AM 9.211). 71 Thus an effect, on
this account, is something which happens to a body as a result of some action of
another body. This ‘something which happens’, however, is not itself a body, but
is something ‘incorporeal’, that is, a predicate. We shall return to the topic of
the predicate and its role in Stoic semantic theory, but for the moment we intend
to give some consideration to the various other types of lekta recognised by the
Stoics.
In the last part of Passage A, Sextus writes that the pragmata which are signified
by the utterance are not only incorporeal, but also spoken or able to be spoken
(Xexxov). They are differentiated first of all between those which are ‘complete’
(auxoxeXrj) and those which are ‘incomplete’ (eXXoif)). Of the complete pragmata,
one is called the axioma , and it is this which is either true or false. In similar
texts strongly suggestive of a common source, Sextus and Diogenes Laertius each
render an account of the various kinds of complete pragmata (AM 8.70-74; DL
7.65-68). In these passages, however, they write of lekta rather than of pragmata
which are spoken or can be spoken. In the text at AM 8.70, Sextus reiterates some
of the things he mentioned in Passage A. In particular, he tells us that the Stoics
maintain in common that the true and the false are in the lekton. He goes on
to report that according to them, the lekton subsists coordinately with a rational
presentation, and that a rational presentation is one in which it is possible that
what is presented be exhibited in discourse. Further on, he also mentions again
that the Stoics call some lekta incomplete, and others, complete. He then provides
a list of several different kinds of complete lekta (AM 8.71-73).
Diogenes Laertius, after relating that Chrysippus takes the subject of dialectic to
be that which signifies (to arjpcavovxoc) and that which is signified (arjpaivopeva),
also reports that the lekton is that which subsists coordinately with a rational
presentation. He provides a brief summary of the doctrine of the lekton , saying
that this theory is arranged under the topic of pragmata and semainomena, and
includes complete lekta, such as axiomata and syllogisms, as well as incomplete
lekta, namely predicates (xaxrjyopripa), both direct (opDcx) and indirect (unxia)
(DL 7.63). 7 ' i He then gives an account of incomplete lekta, or predicates (7.63-65),
70 “As physical events are transmitted by nearby action, either through direct contact of bodies
or by the pneuma, this must be true also for cause-effect relations. Contiguity is therefore an
essential attribute of causality, and causes are bodies acting upon other bodies either in spatial
contact with them or through the medium of the pneuma” (Sambursky [1959, p. 53]).
71 stye cttcmxoi pcv 7iav cuxiov cnopa tpacn acopaxi aawpaxou xivoc aixiov yiveaDai.
72 Hick’s note here is somewhat misleading. He writes that ‘“Direct Predicate’ answers to our
Active Verb, ‘Predicate Reversed’ to our Passive” (DL 7.63 note a). This seems to point out
a fairly fundamental misunderstanding of the concept of a lekton in Stoic semantics. As lekta,
predicates are incorporeal, verbs are parts of speech, and as such, are corporeal. A verb (fjpa),
according to Diogenes the Babylonian, is a part of speech signifying an uncombined predicate
(DL 7.58).
The Megarians and the Stoics
443
which leads into a summary of the various kinds of complete lekta (7.65-68). He
begins this synopsis with the following characterisation of the axioma:
An axioma is that which is either true or false, or a complete pragma
(o(Utot£Xe<; Ttpaypo!) such as can be asserted (dmocpavxov) in itself. Thus
Chrysippus says in his Dialectical Definitions, “An axioma is that such
as can be asserted in itself, as, for example, ‘It is day’, ‘Dion is walking’”
(DL 7.65). 73
There is some question concerning the meaning of ‘dnocpaviov’ in this passage.
Hicks renders it ‘capable of being denied’, but Mates argues that this adjective is
derived from ancxpouvcn, not from aitocpdaxco or dxocprjpi, and so should be translated
as ‘asserted’ or ‘capable of being asserted’ [Mates, 1953, p. 28]. In accord with
Mates’ view, Diogenes writes, just after the text quoted above, that the term
‘dt^iw|jia’ is derived from the verb ‘d^ioucrdaL’. This has the meaning ‘to be asserted’
or ‘to be claimed’. 74
As for the other kinds of complete lekta, both authors present a similar cata¬
logue. There are discrepancies, however, inasmuch as some kinds of lekta appear
on one list but not on the other, and inasmuch as some kinds are not denoted by the
same terminology on both lists. What we report here is inclusive of both lists and
ignores the differences in terminology. This comprehensive list includes questions
of two kinds: interrogations (epwiripaxa), i.e., those which require only a ‘yes’ or
‘no’ reply, and inquiries (xoapaxa), i.e., those which require an explanatory reply.
It also includes imperatives (xpooxoraxoi), prayers (cOxiixai) and curses (dpcxxixcu),
oaths (opxtxoi), hypothetical (ututOetixch), vocatives (Ttpocrayopsimxot), declara¬
tives (dirocpavxixat), a kind of rhetorical question (eKO<Tiopr]xixot), and finally, a
lekton which Diogenes calls a quasi-axioma (opoiov d^tcopct) (DL 7.67). For exam¬
ple, although the sentence ‘Priam’s sons are like the cowherd’ signifies an axioma,
the sentence ‘How like to Priam’s sons the cowherd is!’ signifies a quasi-axioma-,
or, as Sextus puts it, “something more than an axioma, but not an axioma ” (AM
8.73). We have noted that these texts would seem to suggest that the type of lekton
is determined by the type of speech act which is its signifier, and it is tempting
to conjecture that there is some parallel with the speech act theories of Searle,
Hare, and Austin. At any rate, we shall say more about this possibility in the
next subsection.
Not much textual information has come down to us concerning the Stoic treat¬
ment of these various kinds of lekta other than axiomata. We do know, however,
that Chrysippus had an interest in developing a theory of at least some of them,
73 A?iu[ia 8e saxiv o ecmv dXiqttec rj 4 iE v8oc rj npayna auxoxE Xtc ouiorj>avx6v oaov eq)’ eauxG, (Ik
6 Xpuaimmoc cpr)aiv ev xou; AiaXexxixoTc opoic ‘d^iopd eaxi to amocpavxov rj xaxacpavxov oaov ccp’
eauxG)\ Compare the definition given by Sextus at PH 2.104: ‘xal to pev acpcopa cpaaiv eTvcu
Xexxov auxoxeXec; dmocpavxov oaov eqf eauxfiS’. These descriptions differ on in that Sextus has
‘lekton 1 instead of 1 pragma'. And Aulus Gellius reports that he found this definition of the
axioma in his Greek books: ‘Xexxov auxoxeXec dmocpavxov oaov ecp’ auTO)’.
74 For further discussion of the difficulties in rendering this passage, see Frede [1974, pp. 38-40];
Long and Sedley [1990, 2.204-05]; and Margaret Reesor [1989, pp. 46 48].
444
Robert R. O’Toole and Raymond E. Jennings
for he is reported by Diogenes Laertius to have written a series of books on imper¬
atives and questions under the general heading of Logical Topics Concerning Lekta
(DL 7.191); moreover, in a damaged papyrus discovered at Herculaneum we have a
discussion by Chrysippus on the relationship between predicates, statements, and
imperatives.^Nevertheless, aside from these examples and some isolated entries
in the texts of a few commentators “most of the attention was given to proposi¬
tions” [Frede, 1987b, p. 345], and virtually all of the extant writing by the ancient
commentators on the topic of the lekton is about the axioma.
5.2 Lekta and rational presentations
In the passage from Sextus Empiricus quoted above (AM 8.70), we are told that
“the lekton is that which subsists in accordance with a rational presentation” and
that “a rational presentation is one in which it is possible that what is presented
(to cpotvxacrdev) be set forth in language.” Another passage, this one from Diogenes
Laertius, informs us that “those presentations which are rational are processes of
thought” 76 (DL 7.51). If one compares these texts with the text of Passage A, where
Sextus implies that the pragma is that which subsists coordinately with thought,
this comparison would seem to indicate that there is some sort of correspondence
between the terms ‘pragma’ and ‘ lekton ’ in these contexts. Confirmation for this
correspondence is provided by a comparison of the discussion at the end of Pas¬
sage A with what Sextus reports at AM 8.70 concerning complete and incomplete
lekta. In Passage A, Sextus writes that the pragmata which are spoken or can
be spoken (npaypaia Xexxd) differ inasmuch as some are complete, whereas oth¬
ers are incomplete. At AM 8.70, however, he writes that lekta are differentiated
in that some are complete while others are incomplete. What would seem to be
the case, then, is that lekta are pragmata which are spoken or can be spoken.
If this judgment is correct, then it is evident that in the text at AM 8.70 one
could substitute ‘ pragma which is spoken or can be spoken’ for ‘ lekton ’ and thus
interpret Sextus’ remark as ‘the pragma which is spoken or can be spoken is that
which subsists coordinately with a rational presentation’. It also seems plausible
that in the kinds of contexts under consideration the qualifying phrase ‘spoken or
capable of being spoken’ could be dropped. This possibility seems to be realised,
for example, in Passage A, where the pragmata referred to are semainomena, and
hence, one would suppose, spoken. At any rate, by dropping the qualifying phrase,
and substituting just the term ‘ pragma ’ for the term ‘lekton’ in the text at AM
8.70, one could simply say that ‘the pragma is that which subsists coordinately
with a rational presentation’. However, ‘ pragma ’ should always be understood as
‘pragma spoken’ or ‘ pragma which can be spoken’.
75 This book, called Logical Inquiries (AOF1KPN ZHTHMATS7N), is included in the collection
by von Arnim as Fragment 2.298a. For an interesting view on the content of these writings and on
the possible similarity of the Stoic theory to those of modern theorists in the logic of imperatives:
see Inwood’s Ethics and Human Action in Early Stoicism, [inwood, 1985]
76 a! pev ouv Xoyixcd vorjaesc Etcnv.
The Megarians and the Stoics
445
Possibly a relevant text here is that of Diogenes Laertius in which he writes that
under the general heading of dialectic, the doctrine of the lekton is arranged under
the topic of pragmata and semainomena (7.63). These remarks seem to indicate
that lekta belong to a differentia of pragmata and of semainomena. That lekta are
a species of semainomena seems unproblematic, and we shall argue in the sequel
that not all semainomena are lekta. It is not clear, on the other hand, what the
other differentiae of pragmata would be. However, given the understanding of
‘pragmata ’ as ‘states of affairs’, it is plausible to suppose that the Stoics did not
think that all states of affairs are spoken or capable of being spoken.
Another section of the text at AM 8.70 to consider is the remark that a rational
presentation is one for which it is possible that what is presented (to cpavTOtaffcv)
be exhibited by means of discourse. There is some controversy concerning the
interpretation of ‘to cpavTaaffev’ in this passage. Mates writes that ‘to cpavTaaffEv’
is “the objective content of the presentation,” and equates it with ‘to Xextov’
(1953, 22). In criticising this view, Long says that ‘“the presented object’ (sc. to
(pavTaaffcv) is what a phantasia reveals, a ‘thing’ not a lekton. If I see Cato walking
I am presented with an object which can be denoted in a complete lekton ” [Long,
1971, 109m33]. We would reply that what I am presented with is not merely Cato,
but Cato walking, which we take to be not merely an object, but a state of affairs.
We would agree with Long, however, that to Xextov should not be equated with
to cpavTaaffcv. It seems to us that ‘to cpavTacnlev’ should be understood simply as
‘that which is presented’, and the text should be taken as setting out a necessary
and sufficient condition for a presentation to be rational. On this reading, we take
it that the condition is fulfilled if and only if to cpavTaaffev is a itpaypa Xextov ,
that is, a state of affairs which is spoken or can be spoken. On the other hand, the
condition is not fulfilled if to (pavTaaffcv is not a rcpoiypa Xextov.' 7 At any rate,
we take it that Sextus’ remark can be rephrased thus: “A rational presentation is
one in which what is presented is a pragma which is spoken or can be spoken.”
The following quotation from Diogenes Laertius indicates that the Stoics appear
to have drawn a distinction which is of relevance to the present discussion. They
seem to have differentiated discourse from both mere utterance and speech, and
to have referred to discourse as ‘speaking pragmata ’.
Mere utterance (cpcovrj) and speech (Xe^u;) differ inasmuch as mere ut¬
terance is sometimes just noise, whereas speech is always articulate
(svapffpoc;). And speech also differs from discourse (Xoyo<;) inasmuch
as discourse is always significant (oiqpavTtxot;); hence, though speech
(lexis) is sometimes meaningless, as for instance the word ‘pXfrupi’, dis¬
course is never so. And discourse or ‘speaking’ (to XcyEiv) also differs
from mere utterance, for whereas sounds are uttered, states of affairs
(xpayporra) are spoken: and such states of affairs, in fact, happen to
be lekta (DL 7.57).
77 The presentations of children, for example, are not rational until they have accumulated a
certain stock of conceptions (e.g., see footnote 62, page 433), but since they apparently do have
presentations, surely one can speak of ‘that which is presented’ in such presentations.
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Robert R. O’Toole and Raymond E. Jennings
This notion of ‘discourse’ as ‘speaking pragmata' is elaborated by Sextus Empir¬
icus in a passage in which he is reports that “to speak (to Xeyav), according to
the Stoics themselves, is to utter sounds capable of signifying the state of affairs
(xpaypa) apprehended” (AM 8.80). Thus one might say that to engage in dis¬
course, that is, to speak pragmata, is to utter articulate sounds which signify the
state of affairs apprehended in a rational presentation.
Other passages which seem relevant are those recorded by Diogenes
Laertius at 7.66-67. Here he informs us that, according the Stoics, an axioma
“is a [state of affairs] which we assert to be the case when we speak it” (7.66). 78 we
take it that what this means is that when one ‘speaks a pragma ' 79 by asserting
it, the lekton , that is, what is said, is an axioma: to put it another way, what is
said when one asserts that some state of affairs holds or is the case is an axioma
or proposition. It might be instructive to compare Diogenes’ account of the lekton
called an imperative (TtpoaxaxTixov): “An imperative is a pragma which we com¬
mand to be the case when we speak it” (DL 7.67). 80 Thus, when one speaks a
pragma by commanding it, the lekton , or what is said, is a prostaktikon: in other
words, what is said when one commands that some state of affairs be the case is a
prostaktikon or imperative. Similarly, when one speaks a pragma by asking it, the
lekton is a query or interrogation (epd>Tr)[ia). According to Diogenes, the sentence
‘It is day’ signifies an axioma whereas the sentence ‘Is it day?’ signifies an inter¬
rogation. What this example would seem to indicate is that the same pragma or
state of affairs—in this case, its being day—can function as the content of various
types of lekta, depending on the speech act involved.
Brad Inwood plausibly suggests that this conception is comparable to the idea
familiar in the speech act theories of Searle and Hare, that is, the idea of a “dis¬
tinction between content and mode of assertion” [inwood, 1985, p. 93]. 81 It is
similar, for example, to the distinction made by Searle “between the illocution¬
ary act and the propositional content of the illocutionary act” (Searle, [1969, p.
30]). By ‘illocutionary acts’ he means acts of “stating, questioning, commanding,
promising, etc.” [Searle, 1969, p. 24], and by ‘propositional content’ he seems to
mean (in the case of asserting or stating, for example) “what is asserted in the
act of asserting, what is stated in the act of stating” [Searle, 1969, p. 29]. Thus
according to Searle, uttering the sentence ‘Sam smokes habitually’ constitutes the
performance of a different illocutionary act than uttering the sentence ‘Does Sam
78 od;u3pa Y&P eaxiv [xpaypa] 6 Xsyovxec; ditoqjaivotiE'Oa. My justification for inserting
‘itpaypa’ into the text here is twofold: first, there is the passage at 7.65 where Diogenes de¬
scribes the axioma as a ‘xpaypa auxoxXec axotpavxov oaov e<p’ cauxto’; second, in the passages
from 7.66-68, there are the instances of ‘xpaypa’ occurring in similar grammatical constructions
in the descriptions of the other types of lekta. For the translation in these contexts of ‘npaypa’
as ‘state of affairs’, see Long, ([1971], 107nl0; Long and Sedley, [1990], 1.195, 202; Reesor, [1989,
Ch. 3],
79 See Margaret Reesor’s comments on the Stoic notion of speaking as ‘speaking a pragma’
(state of affairs) [Reesor, 1989, pp. 33-34].
80 7tpocrxctxxix6v 8 e eoxi xpaypa o Xeyovxei; npocraxoaopEV.
81 As it is used in the phrase ‘mode of assertion’, the term ‘assertion’ should be understood as
neutral among the various illocutionary acts.
The Megarians and the Stoics
447
smoke habitually?’ or ‘Sam, smoke habitually!’, but although the ‘mode of asser¬
tion’ differs in each case, the propositional content, which might be represented
by the complex {Sam, smokes habitually}, is the same. Inwood has reservations
about this comparison, however, writing that “for the analogy to a speech act
theory like Hare’s or Searle’s to be complete, it would have to be the case that the
Stoics isolated a subject-predicate complex from its mode of assertion. And they
appear not to have done this” ([inwood, 1985], 95). We shall attempt to develop
an interpretation to the contrary, an interpretation in which the complete pragma
(to auioxeXec; jipaypa) is just such a complex.
At 7.49, Diogenes Laertius details an order of priority between presentation
and discourse. He writes that according to the Stoics, “presentation is first, then
thought, which is capable of speaking out, discloses by means of discourse that
which is experienced through the presentation.” At 7.57, Diogenes also writes that
pragmata are spoken and that discourse is to speak pragmata. If one interprets
‘to Xextov’ to mean ‘that which is spoken or can be spoken’, an interpretation
we shall argue for in the sequel, then one can render the text at AM 8.70 as
‘that which is spoken or can be spoken is that subsisting coordinately with a
rational presentation’. Since ‘that which is spoken’ is the pragma , the foregoing
interpretation becomes ‘the pragma is that subsisting coordinately with a rational
presentation’. Hence, the pragma will also be prior to discourse, since it subsists
coordinately with the rational presentation. And of course, if the pragma is what
is spoken, it would seem to be prior to discourse in any case. This priority, we
take it, along with the passage at DL 7.66-67 in which Diogenes characterises each
type of lekton as a pragma spoken in a certain mode, is a strong indication that
the Stoics isolated the pragma from its ‘mode of assertion’.
The question arises as to the nature of the pragma , or more particularly, of the
complete pragma. The simplest procedure for setting out an account of this item
will be to consider an example of a sensory presentation which is a presentation
of a real feature of the world. A general characterisation of the pragma might be
that it is an abstract structure which is the result of a mental process whereby the
mind interprets the actual state of affairs apprehended in the presentation. This
structure is assembled from the appropriate conceptions selected from those stored
in the mind’s stock of conceptions, 82 and it is this interpretation which allows what
is perceived to be represented in language. Pragmata , then, are abstract structures
which correspond, on the one hand, to the language used to represent them, and
on the other hand, to the actual states of affairs or situations which engender the
presentations of which they are the content. This latter correspondence, however,
only holds in the case of veridical presentations. 83
82 See page 433 for a discussion of how rationality is completed from our preconceptions
(npoXrj^Etc;), and see page 434 for a discussion of how more complex conceptions might be pro¬
duced from these primary conceptions.
83 Contrast Graeser, [1978b, p. 8]: “[The Stoics] insisted that there holds no isomorphic cor¬
relation between thought on the one hand and things-that-are on the other. ... [They] implied
that ontological analysis is bound to be subjective, or rather functional, in that it is man’s mind
that superimposes its concepts on reality.” It seems to us, however, that if someone believed that
448
Robert R. O’Toole and Raymond E. Jennings
Consider an example. Suppose we see Dion walking. What there is, according
to the Stoics, and hence, what is perceived, is the individually qualified substance
of Dion in a certain state (mo<; s/ov), that is, possessing the attribute ‘walking’ (to
TrepiKaxElv). 84 The mind searches its stock of conceptions, and if the conception of
Dion and of the attribute ‘walking’ are among them, then it possesses the necessary
components for constructing the pragma. In general, for a simple example such
as this, the components of the pragma could be thought of as an ordered pair
of items, of which the first is either an individuating quality (iSia tcoiott]?) or a
common quality (xotvf) jioidTTjt;), and second is a predicate (xcmrjyoprjjia). For a
particular pragma , the first of these components would be signified by either an
individual name or a common name, and the second, by a nominalised infinitive
verb. For convenience, we shall represent such a structure by first writing down a
left brace, then the name signifying the individuating quality, then a comma, then
the nominalised infinitive verb signifying the predicate (AM 9.211; Clement strom.
8.9.30.3), and last, a right brace. Hence, for the example under consideration, the
pragma will be represented thus: {Aitov, to TteputaTel}. In English this will be:
{Dion, to walk}. 85
We take those passages at AM 8.80 and DL 7.66-67 to indicate that to ‘speak a
pragma ’ is to perform what Searle defines as an ‘illocutionary act’ (24). According
to these texts, the result of speaking a pragma in a certain mode is a certain type
the same logos which structures reality is also immanent in our minds, then one would expect
them also to believe in some sort of isomorphism between thought and reality. But then Graeser
seems to take what Long and Sedley refer to as a “variant reading” [Long and Sedley, 1990,
1.274] of Diogenes Laertius 7.134 as evidence that the logos itself is incorporeal [Graeser, 1978b,
p. 99]. He mentions Posidonius as possibly holding such a view [Graeser, 1978b, p. 99]. This
may be so. However, as Long and Sedley have argued [1990, 1.274], this cannot be the view of
the Old Stoa.
84 According to Stobaeus, Chrysippus held that even predicates are real, but only those which
are actual attributes (aup(3s(3r|x6Ta). He says that “‘walking’ (to TtepntaTElv) is real (uTtapxeiv)
for me when I am walking, but it is not real when 1 have lain down or am sitting down” ( eclog.
1.106.18-20).
Seneca records a dispute between Kleanthes and his pupil Chrysippus about the nature of
walking. According to Kleanthes walking is breath (spiritus = pneuma) extending from the
he.gtmon.ikon (principalis = hegemonikon) to the feet, whereas, according to Chrysippus, it is
the hegemonikon itself ( epist. 113.23). Leaving aside the question of how the dispute turned
out, it seems apparent that whatever else they meant by the term ‘walking’, both Kleanthes and
Chrysippus thought that they were talking about something corporeal, for in Stoic doctrine both
the pneuma and the hegemonikon are bodies (SVF 2.879). But since there is no doubt that they
conceived of predicates as being incorporeal, they clearly could not have been referring to the
predicate {walking} by their use in this context of the term ‘walking’.
85 It seems apparent that the Stoics used the nominalised infinitive verb to signify a predicate.
For example, at AM 9.211, Sextus reports that according to the Stoics, “the scalpel is corporeal,
and the flesh is corporeal, but the predicate ‘to be cut’ is incorporeal” (aiopa pev to crpiXiov,
CKopart 8e xfj aapxi, aaupaxou 8 e tou Tepveaflai xanriyopiQpaToc) (cf. Clement strom. 8.9.30.3).
Although it may seem more natural to render the nominalised infinitive by a gerund, for example,
‘walking’ rather than ‘to walk’, it is not always the simplest representation, particularly in the
case of complex predicates. Note that a finite verb seems to be the signification of an incomplete
predicate. So at 7.63 Diogenes Laertius says that the verb ‘ypoKpEi’ (He/she writes) signifies an
incomplete predicate (see the next section for further discussion of incomplete predicates).
The Megarians and the Stoics
449
of lekton. Obviously, ‘ lekton ’ (or, more strictly, ‘Ttp&YP°! kexiov’) will have the
sense in these contexts of ‘pragma spoken’. An axioma , on this account, is the
result of speaking a pragma by asserting it, a prostaktikon is the result of speaking
a pragma by commanding it, and a similar account can be given for the other
types of lekta. Moreover, an axioma is what is asserted in the act of asserting,
a prostaktikon is what is commanded in the act of commanding, and so on. One
can probably think of the lekton as an abstract structure which will include the
elements of the associated pragma , but which will have a richer structure in that
it will contain items not part of the pragma. For example, lekta will obviously
have moods, and probably tenses as well. At any rate, as the Kneales point out,
axiomata will have tenses [Kneale and Kneale, 1962a, p. 153]. There may be
items corresponding to various sentence operators, such as operators for negations
and questions. In addition, there may be items corresponding to connectives and
articles (cf. DL 7.58). Although We will need to look at axiomata which involve
items corresponding to connectives, we do not intend to give an analysis of the
structure of lekta in general; hence, for the most part, we will simply represent
a lekton by writing down its signifying sentence and enclosing it with a pair of
braces. For example, one way in which one could speak the pragma Dion, walking
would be to utter the sentence ‘Dion is walking’. The axioma associated with this
utterance would be represented thus: {Dion is walking}.
At this point, there are some observations which should be made. First, it is
evident that the pragma is what might be called the ‘propositional content’ of a
rational presentation, but we would as soon avoid using this expression. Some
commentators who speak of the ‘propositional content’ of a rational presentation
seem to suppose that this content is a proposition (e.g., Frede [l987e, p. 154]).
However, the only item which could be compared to a proposition in Stoic seman¬
tics is the axioma, and we do not see why, supposing that the content of a rational
presentation is a lekton , it should be an axioma rather than some other type of
lekton. Second, the formation or construction of the pragma would appear to be
a constituent of the perceptual process. According to Chrysippus, a presentation
reveals itself and that which caused it (AM 7.230; Aetius plac. 4.12.1, DDG 401).
Thus one is conscious of the mental process which is the presentation, as well as
the external state of affairs (in the case of a sensory presentation) which caused
the presentation. Sandbach writes that the presentation thus “gives information
about the external object” [Sandbach, 1971b, p. 13]. But clearly, without the
pragma , which we take to be the mind’s interpretation of the external state of
affairs, there can be no information received, and hence, no perception. Third, it
was suggested above that a presentation is rational if and only if there is a pragma
subsisting coordinately with it. This would be a lekton in the sense of a ‘ pragma
which can be spoken’. We do not think that there is necessarily a lekton subsisting
coordinately with the presentation in the sense of a pragma spoken. This result
would seem to be indicated by the priority of the presentation with respect to
discourse (see page 428).
450
Robert R. O’Toole and Raymond E. Jennings
It is clear that this account of the lekton is fairly rudimentary at best. For
example, we have said nothing of how this interpretation will function for non-
sensory presentations. Although we will need to address this topic in particular
and expand certain other aspects of the account as well (aspects such as axidmata
involving connectives, already mentioned above), we believe that what has been
said so far will serve as a basis for developing a characterisation of the role of the
axioma in the theory of inference.
5.3 Incomplete lekta
Nouns and incomplete lekta
Another relevant point not brought out in Passage A but mentioned just after,
is the distinction among lekta drawn by the Stoics between those lekta which are
‘complete’ (auxoxeXet;) and those which are ‘incomplete’ (eXXltccc;). This distinc¬
tion is confirmed by Sextus in another passage (AM 8.70-74) and also by Diogenes
(7.65-68). Incomplete lekta, according to Diogenes, are those for which the signi¬
fying expression is also incomplete. For example, ‘He writes’ (rpacpei), although
a grammatically complete expression, signifies an incomplete lekton, presumably
because it lacks a definite subject, and hence, does not signify a complete state
of affairs. A complete lekton, on the other hand, is one signified by a complete
expression, for example, ‘Socrates writes’ (rpacpei £caxpdxr)<;) (7.63). At 7.58 Dio¬
genes reports that a verb (fjpa) signifies an uncombined predicate, and at 7.64
he gives a characterisation of a predicate (xaxfiyopf|[ia) as “an incomplete lekton
which has to be combined with a nominative case (opffo c xxwok;) in order to form
a complete lekton .” Given that the expression ‘Socrates writes’ signifies a complete
lekton these two passages would seem to suggest that the significatum of a noun
such as ‘Socrates’ occurring in the subject position of a sentence such ‘Socrates
writes’ is, according to the Stoics, a nominative case (opfioc irxwaic;) (DL 63-64).
Moreover, no matter how odd or even obscure it might seem to us, what is further
suggested is that for the Stoics the cases (hai ptoseis) are not understood primarily
in a grammatical sense.
From what has been said above, it is clear that an isolated verb such as ‘writes’
can signify an incomplete lekton. An issue which arises is the question whether
isolated nouns can also signify incomplete lekta. Many commentators seem to think
that they can, 86 and they seem to think so for one or both of two closely connected
reasons. One reason is the example given by Sextus Empiricus in Passage A.
Recall that Sextus informs us in this passage that the Stoics distinguish among
three things: that which signifies (to semainon), i.e., the utterance (he phone);
that which is signified (to semainomenon), i.e., the lekton; and the subject of
predication (to tynchanon), i.e., the external existent (to ektos hypokeimenon).
As an instance of that which signifies he cites the utterance ‘Dion’. Given this
86 Mates [1953], 16-17; Kneale & Kneale [1962a], 144, 148; Watson [1966], 47-49; Graeser
[1978b], 91; Sandbach [1975], 96.
The Megarians and the Stoics
451
example it seems natural to suppose that there is a lekton associated with this
expression and that one may take this lekton to be something like its sense or
meaning; one may suppose, moreover, that the referent of this meaning is the
object picked out, i.e., Dion himself. All this seems to suggest a Fregean semantic
analysis of the lekton, and this is the course which some authors appear to take. 87
There is a difficulty with this approach, however, and it involves the fact that
in Passage A Sextus is giving an account of a controversy over what it is that is
true or false. According to him, the Stoics locate truth and falsity in ‘that which
is signified’, which, as we have seen, is the axioma. But we have also seen that an
axioma is signified by the utterance of a declarative sentence. Hence one would
expect that Sextus would give a declarative sentence as an example of an utterance
which is ‘that which signifies’. Whatever else it might be, the utterance ‘Dion’
seems clearly not to be the utterance of a declarative sentence. Hence it is not the
significans of an axioma, and thus not the signiftcans of anything either true or
false. The inappropriateness of Sextus’s example is emphasised by consideration
of a passage from Seneca’s Epistulae Morales. The content of this passage would
seem to parallel that of Passage A.
I see Cato walking. The sense (of sight) reveals this (state of affairs),
the mind believes it. What I see is an object, toward which I direct
both (my) sight and (my) mind. Then I say: “Cato is walking.” What
I say now, according to them, is not an object, but something declar¬
ative about an object: this (that I say) some call ‘effatum’, others
‘enuntiatum’, and others ‘dictum’. 88 Thus when we say ‘wisdom’ we
understand something material; when we say ‘He is wise’, we say (some¬
thing) about an object. It makes a great deal of difference, therefore,
whether you indicate the object or say something about it (3: 117.13).
It is apparent that something like Seneca’s example ‘Cato is walking’ is needed,
and this requirement is all the more apparent when one considers the examples
given by Chrysippus as quoted by Diogenes Laertius, i.e., ‘It is day’, ‘Dion is
walking’ (DL 7.65). One proposal for clearing up this problem is the suggestion
that uttering ‘Dion’ may be taken as “equivalent to asserting the true proposition
‘this man is Dion’” [Long, 1971, p. 77, 107nll], Whatever are the merits of this
particular suggestion, it seems that something of this sort must be posited, for
we have another passage similar in context to Passage A in which Sextus also
mentions the expression ‘Dion’ as being the significans of an “incorporeal lekton“
(AM 8.75). Hence we cannot simply write off the example as an aberration in
Sextus’s account (cf. Frede [1987b, p. 349]). Be that as it may, we think that
the infelicity of Sextus’s example for the point it is meant to illustrate renders it
questionable as evidence that Stoics viewed isolated nouns as significantia of lekta.
87 For example Mates [1953, p. 19]; Gould [1970, 70nl],
88 Note that ‘effatum’, ‘enuntiatum’, and ‘dictum’ are Latin translations of &!;t'o>pc((cf. Cicero
acad. 2.95 [effatum]; de fato 19.28 [enuntiatum = enuntiatio}).
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Robert R. O’Toole and Raymond E. Jennings
Recall that, according to Passage A, the Stoics suppose that the sign (to semainon
and the object of reference (to tynchanon) are both corporeal (aojpaxixov), whereas
the lekton is incorporeal (dawfionroc)- We wish to discuss an assumption made by
some authors which is based on the posited immaterial nature of the lekton. This
assumption leads to the second reason for an affirmative reply to the question
whether isolated nouns can signify an incomplete lekton. This is the assumption
that since the lekton is incorporeal, whatever are its constituents must also be
incorporeal. Hence, since the significatum of a noun such as ‘Socrates’ can be
a constituent of a complete lekton —for example, the lekton signified by the ex¬
pression ‘Socrates walks’—it would seem to follow that the significatum of the
expression ‘Socrates’ is incorporeal. But if it is, then it must be a lekton of some
sort, since it could hardly be an incorporeal belonging to any one of the other
classes of immaterial entities. Now it seems clear that the expression ‘Socrates’
will not signify a complete lekton; therefore, it seems natural to conclude that this
expression signifies an incomplete lekton, and that, in general, isolated nouns can
signify incomplete lekta.
Given the Stoic view cited above that nouns signify cases (xxtoaeic;), it would
seem to follow from the argument in the last chapter that a case is an incomplete
lekton and hence something incorporeal. On the other hand, if one assumes that
“the ptosis is definitely conceived of as something incorporeal” [Graeser, 1978b,
p. 91], then it would seem to follow that a case is an incomplete lekton. Either
way we get the conclusion that isolated nouns can signify incomplete lekta. One
attractive feature of this argument is that it fits in rather well with the example
‘Dion’ presented by Sextus Empiricus in Passage A; indeed, some writers conclude
that the supposed incorporeal nature of the cases provides confirmation of the le¬
gitimacy of Sextus’s example (cf. Graeser [1978b, p. 9lj), whereas others conclude
that Sextus’s example provides confirmation that the Stoics viewed the cases as
incorporeal. 89
We have already suggested that Sextus’s example is suspect as evidence that
isolated nouns signify lekta, and this would seem to count against the view that the
Stoics thought of the cases as incorporeal. However, there are two other objections
to these theses which would seem to be somewhat stronger. The first is based
on the fact that in any discussion of this subject in the sources, only predicates
are ever mentioned as being incomplete lekta (cf. Frede [1987b, p. 347]; Long
[1971, pp. 104-05]; Graeser [1978b, p. 91]). The other is based on the report of
Diogenes Laertius (7.58) to the effect that the Stoics assumed that the significata
of names and common nouns are, respectively, individual qualities (tSioti TtoibxrjiEc)
and common qualities (xoivai xoioxrjxec;). Now there is no doubt that the Stoics
assumed that the qualities of material objects were themselves material; 90 hence,
if proper nouns and common nouns signify qualities, and qualities are corporeal,
89 These conclusions are discussed both by Frede [1987b, p. 349] and by Long [1971, p. 105];
however, neither author agrees with them.
90 cf. Rist [1969a], 159; Long [1971], 105; Frede [1987b], 347. For citations from the sources see
SVF 2.449, 463; DL 7.134; Simplicius in cat. 209.10.
The Megarians and the Stoics
453
there would seem to be a difficulty for anyone wishing to maintain the view either
that nouns signify incomplete lekta , or that cases are incorporeal. 91
One might suppose that this should resolve the matter, but at least two more
complications arise. The first complication involves two passages in Clement of
Alexandria’s Stromateis in which it is claimed that “a case is incorporeal ... and
... agreed to be incorporeal” (Frede [1987b], 350). As for the claim that a case
is incorporeal, Frede has argued convincingly that non-Stoics of later periods in
Greek philosophy would use the term ‘case’ with the conviction that cases are
incorporeal “because they did not share the Stoic view that qualities are bodies”
([Frede, 1987b], 350). As for the claim that cases are agreed to be incorporeal, he
argues that the examples cited by Clement are examples of things which would no
doubt be agreed to be incorporeal by the Stoics, but which would not be agreed
by them to be examples of cases. So much then for the difficulties raised by the
passages of Clement.
The second complication involves interpreting a passage of Stobaeus (SVF 1.65)
in such a way that common qualities, at least, are shown to be incorporeal (cf.
Rist [1969a, p. 165]). This passage, which is described by Frede as “notoriously
obscure and difficult” [Frede, 1987b, p. 348]), is as follows:
Zeno <and his followers > say that concepts (evvofjpcaa) are neither
somethings (itva) nor qualified things (xotot), but are mere images in the
mind—only quasi-somethings or pseudo-qualified things (cboavri 8e uva
xcd cbcravd Ttoia). These (sc. concepts) are called ideas by the ancients.
For the ideas are (ideas) of the things falling under (uTtoitnrcovTcov) the
concepts, such as of men, or of horses, or, speaking more generally,
of all living things, and of any other things which they say are ideas.
The Stoic philosophers say that these (sc. concepts) are non-existents,
and that whereas we participate (pcxEyciv) in the concepts, the cases,
which they call prosegoria , we possess (xuyxdvav).
Following Frede [1987b, pp. 348-49], we take it that the substance of this passage
is claim that the Stoics from Zeno on refused to grant the Platonic Forms or Ideas,
which they called ‘concepts’, any existential status at all—not even the existential
status of the incorporeals such as lekta, void, place, and time. However, there are
things which, because they ‘fall under’ (ukotujiteiv) the concepts, are called ‘cases’
(ktucteic) by the Stoics, and which are contrasted with the concepts, the contrast
91 Rist outlines this difficulty as follows:
Our problem is why the Stoics put these common qualities into the category of
quality, that is, of material objects ... rather than with other incorporeals like
time, void, place and the lekta. The answer to this is not easy to find [Rist, 1969a,
pp. 165-66].
It should be noted that the purported textual evidence adduced by Rist and others to show
that common qualities are not corporeal, would not show, even if it were correct, that common
qualities were classed with the incorporeals such as lekta. What would be shown, as we soon
shall see, is that, common qualities had no ontological status at all.
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Robert R. O’Toole and Raymond E. Jennings
being that corporeal objects merely participate in the latter, whereas the cases
they possess.
Now according to Frede [1987b, p. 348], the Platonists assumed that in ad¬
dition to the transcendental forms or ideas, there are immanent forms which are
embodied in concrete particulars. He suggests that the immanent forms of the
Platonists correspond to the Aristotelian forms, and that both are qualities of
some kind. Thus the Platonists differentiate between the transcendental form wis¬
dom, which the Stoics would call the concept wisdom, and the embodiment of
the form in Socrates himself, i.e., Socrates’ wisdom. On the Stoic view, according
to Stobaeus, Socrates’ wisdom would be a case (ptosis), because it falls under
(hypopiptein) the concept wisdom. Moreover, Socrates would merely participate
in the transcendental form wisdom, whereas he would possess the embodiment of
that form. Hence the Stoic cases appear to correspond to the embodied forms of
the Platonists and the Aristotelians, and like them, appear to be qualities of some
kind. As such, they would be corporeal on the Stoic view, although they would
be incorporeal on the Platonic or Aristotelian conception.
This interpretation of Stobaeus’s passage seems to us to capture the substance
of the Greek. Unfortunately, not all commentators agree. Rist, for example, thinks
that we can deduce from this passage that it is the common qualities of the Stoics
which correspond to the Platonic Forms, and hence, that such qualities must have
been given the same ontological status as the Forms—which is to say, they were
thought of as non-existents ([Rist, 1969a, p. 165]; cf. Reesor [1954, p. 52]). It is
not difficult to see how one might arrive at this interpretation, for there would seem
to be some confusion created inasmuch as the common nouns such as ‘man’ and
‘horse’, which, on the Stoic view, signify common qualities (DL 7.58), are used by
the commentators to refer to the ideas or concepts. Thus, since the term ‘horse’ is
used to talk about both the quality common to all horses and the concept ‘horse’,
it should not be surprising that the concept and the common quality are taken to
be identical. Now it may be that the Stoics themselves are to blame for at least
some of this confusion, for it is easy enough to be careless about the distinctions
one draws. On the other hand, given the view that the cases are qualities, and
given the distinction between cases and concepts—both of which are integral to
Frede’s interpretation of Stobaeus’s passage—there is no reason to suppose that the
Stoics did not intend to maintain the distinction between concepts and common
qualities. But if this distinction is observed, then there would seem to be no basis
for maintaining that common qualities are not corporeal.
It should be noted that the distinction between concepts and cases is mentioned
in other ancient texts. Simplicius, for example, using language similar to that of
Stobaeus, reports that the Stoics called the concepts ‘pcdexid’ (which may be
translated as particibilia (cf. Frede [1987b, p. 348]), because they are participated
in (pexexea'dai) and the cases ‘possessibles’, because they are possessed (Tuy^aveiv)
(in cat. 209.12-14).
The Megarians and the Stoics
455
Predicates
At 7.63 Diogenes Laertius comments that the class of incomplete lekta includes all
predicates (xaxrjyopVjpotxa) and if we are correct in rejecting the significata of iso¬
lated nouns from this class, then it includes only predicates. A predicate, according
to Diogenes, is “that which is said about something, or a pragma constructed from
one or more elements, or (as we have already noted above) an incomplete lekton
which must be joined on to an nominative case (opfloc; irxwaic;) in order to yield
an axidma ” (DL 7.64). The first two of these characterisations are attributed to
the Stoic Apollodorus and the passage is translated by Hicks as if they were in
conflict with the last one. But it seems to us that the versions of Apollodorus are
compatible with the third one and that the three are merely alternatives. Given
what has been said above about complete lekta , we take the sense of the first of
these alternatives to be the idea that in order to signify a complete lekton one
must make some attribution to an object, and what is attributed is a property or
attribute. Corresponding to this property at the level of lekta, is a predicate or
incomplete pragma , and, at the level of language, a verb. In the example of the
above paragraph, writing has been asserted about Socrates to form the axidma
‘Socrates writes’. But it would seem that one could also form the interrogative
(epcoxrjpa) ‘Is Socrates writing?’ by asking of Socrates whether he is writing (cf.
DL 7.66). In either case we take it that {ypoKpei} is ‘that which is said’ about
Socrates.
The second alternative seems to reflect an ambiguity in the Stoic use of the term
kategorema which we are rendering as ‘predicate’. This ambiguity has been noted
by Michael Frede in his article “The Origins of Traditional Grammar” [Frede,
1987b, pp. 338-59]. According to Frede, the Stoics made a distinction between
those predicates which are simple and those which are complex [Frede, 1987b, p.
346]. There seems to be good reason to take the latter to be the result of combining
a ‘direct predicate’ (to opflov xaxr)yopf|pa) (DL 7.64) with an oblique case (f) opflf)
urGcnc). Such complex predicates, according to Diogenes, must be constructed
in this way so as to be capable of combining with a nominative case to produce
a complete lekton. The examples cited by Diogenes are signified by verbs such
as ‘hears’, ‘sees’, and ‘converses’, and these are contrasted with those signified by
such verbs as ‘thinks’ and ‘walks’ (DL 7.64). Now we know that for the Stoics a
verb (fjpa) is “a part of speech signifying an uncombined (aauvdexov) predicate”
(DL 7.58). Hence, it seems to be the case that some uncombined predicates (e.g.,
those cited as instances of direct predicates) cannot as they stand be joined with
other elements to produce a complete lekton , the reason being that there is a sense
in which the verbs signifying such predicates must first be combined with other
parts of speech before they can be joined with a name or common noun in the
nominative case to produce a complete thought. For example, in comparison with
‘Dion is thinking’ or ‘Dion is walking’, it seems plausible that the sentence ‘Dion is
seeing’ or the sentence ‘Dion is hearing’ requires a complement in order to express
a complete thought. To recapitulate, there are, it would seem, some uncombined
456
Robert R. O’Toole and Raymond E. Jennings
predicates which are required to be joined with other elements before they can
partake in the production of a complete lekton. On the other hand, there are
some which can partake in the production of a complete lekton just as they stand.
Frede calls the former ‘syntactic predicates’ and the latter ‘elementary predicates’
([Frede, 1987b], 346). We shall adopt this terminology in the sequel.
Given this ambiguity in the term kategorema , one can think of the third al¬
ternative characterisation of a predicate as a recipe for constructing an axidma
from either an elementary predicate or a syntactic predicate. On the one hand,
an incomplete lekton which is an elementary predicate can be combined with a
nominative case to produce an axidma without further ado. On the other hand,
an incomplete lekton which is a syntactic predicate must first be combined with
other elements before it can be joined with a nominative case to form an axioma.
It is perhaps appropriate at this point to make explicit what is suggested in
the preceding discussion concerning the ‘construction’ of an axioma: that is, the
notion of a syntax of lekta. 92 A good way to introduce this task is to notice that
the Greek noun from which the English word ‘syntax’ is derived, is itself derived
from the Greek verb (oOvxot^u;) and that various forms of this verb are used by
Diogenes in the passages from 7.63 to 7.74 to indicate the notion of ‘putting
together’ various kinds of elements to form a lekton (cf. Frede [1987c, p. 323,
246]; Elgi, in Brunschwig [1978, p. 137]). Furthermore, we have evidence that
Chrysippus, at least, was interested in such a notion, for among the more that
seven hundred books he is reported to have written (DL 7.180), the following
titles appear: (2) On the Syntax of What is Said, four books (Ilepi ifj<; auvid^ecat;
tov Xeyopevcov S'), and (3) Of the Syntax and Elements of What is Said, to Philip,
three books (Ilepl xfjc; auviot^ewc xai axoixeuov xffiv Xeyopevcov repot; $lXuntov y')
(DL 7.193). 93 Now according to A.A Long, the expression ‘xo Xeyopevov’, which
may be rendered as ‘what is said’ or as ‘that which is said’, is extremely difficult
to distinguish in sense from to lekton ([Long, 1971], 107nl3); moreover, as Frede
points out, the passage at DL 7.57 indicates that ‘what is said’ is in fact a lekton.
These writings of Chrysippus seem to reinforce what is implicit in Diogenes’ report
of the Stoic characterisation of a predicate at DL 7.64 and in the whole discussion
from DL 7.63 to DL 7.74: that is, that the Stoic theory of the lekton included the
conception that lekta were analysable into various elements and that there was
a set of syntactic principles whereby such elements were to be joined together to
form a lekton.
92 Much of what we write on the notion of a syntax of lekta is drawn from two papers by
Michael Frede, both of which appear in his Essays in Ancient Philosophy. These are: “The
Principles of Stoic Grammar” (especially 323-32) and “The Origins of Traditional Grammar”
(especially 353-57).
93 These titles have been numbered as they are since they are second and third in a sequence
of four to which we shall refer.
The Megarians and the Stoics
457
5.4 Lekta and parts of speech
The Stoics seem to have thought that one constructs a lekton, and in particular
an axidma, by combining the elements of the corresponding declarative sentence
in the right way. This amounts to putting together the elements of the sentence
in such a manner that the structure of the corresponding axidma is syntactically
correct, its elements being combined in accordance with the syntax of lekta (cf.
Frede [1987c, p. 324]). The supposition that the Stoics entertained some such
notion of a relation between the elements of a sentence and the elements of the
corresponding lekton is suggested by the remaining two titles in the sequence of
four mentioned above. These are: (1) On the Elements of Speech and on Things
Said, five books (ITepi xcov axoixeftuv too Xoyou xal xwv Xeyopevcov e'), and (4)
On the Elements of Speech, to Nicias, one book (Ilepl xcov ctxolxeiwv xou Xoyou
xpot; Nixlav a) (DL 7.193). Note that (1) and (4) are concerned not only with
the parts of speech but also with the elements of lekta, and that (2) and (3) are
concerned not only with the elements of lekta but also with the syntax of lekta.
The placement of these titles in this particular sequence seems to point to “a
systematic connection between parts of speech, elements of lekta, and the syntax
of lekta ” [Frede, 1987c, p. 324]; moreover, such a connection would be explained
by assuming that the Stoics envisaged the production of a lekton to take place in
accordance with the theory outlined above.
As for the connection between the parts of speech and the elements of lekta, we
have a text of Diogenes Laertius which seems to suggest that this connection is
a relation of signification. Both Chrysippus and Diogenes the Babylonian, 94 ac¬
cording to this text, stated that there are five parts of speech: these are individual
name, common name, verb, conjunction, and article. Diogenes, in his treatise On
Language, associates at least the first three of these with the corresponding ele¬
ments of lekta. An individual name (ovopa), according to him, is a part of speech
indicating (SrjXouv) an individual quality (18ta Koioxrjc) (e.g., Diogenes, Socrates);
whereas a common name is a part of speech signifying (cnipotTvov) a common qual¬
ity (xoivf) TtOLOxrjc) (e.g., man, horse). A verb (fjpa) is a part of speech signifying
an uncombined predicate (as we have already seen). A conjunction (auvSeopoc;)
is an indeclinable part of speech, binding together the parts of a sentence, and
an article (apdpov) is a declinable element of a sentence, determining the genders
and numbers of nouns (DL 7.58). The relation outlined in this passage between
the elements of speech and the elements of lekta seems clear with respect to the
first three parts of speech on the list. It also seems clear what the corresponding
element at the level of lekta is for each of these parts of speech. If we take the
participle ‘SrjXouv’ to mean ‘signified’ in this context, then we can suppose that
for the Stoics there is a relation of signification respectively between individual
names, common names, and verbs at the level of parts of speech, and individual
qualities, common qualities, and predicates at the level of lekta.
94 A Stoic also known as Diogenes of Seleucia, but called The Babylonian because Seleucia is
near Babylon (cf. DL 6.81).
458
Robert R. O’Toole and Raymond E. Jennings
The passage is not so clear, however, about the elements of lekta which are
supposed to correspond to conjunctions or articles. One assumes, given what
has been said about the other three parts of speech, that whatever the nature of
these elements of lekta might be, the connection between each of them and the
corresponding part of speech should also be one of signification. But it is difficult
to tell from the text, for conjunctions and articles are defined grammatically rather
than by their signification at the level of lekta. In addition, as Frede points out
[Frede, 1987c, p. 331], there is a difficulty inasmuch as the Stoics think that the
class of conjunctions includes both conjunctions proper and prepositions, and that
the class of articles includes both articles proper and pronouns. Thus it is not
at all transparent how one is to envisage an element of a lekton which can be
the significate both of conjunctions and of prepositions, or one which can be the
significate both of articles and of pronouns.
5.5 Ontological concerns
In this section we intend to consider briefly some issues concerning the ontological
status of the lekton. The first of these is the problem of how one ought to interpret
the meaning of the term ‘lekton'. The question presents some difficulty inasmuch
as it seems to be connected with ontological concerns. The other topic is the
question whether the lekton was conceived by the Stoics as merely a construct of
the mind, or as something having a more tangible status.
The interpretation of Text ov'
The substantive expression ‘to Xcxtov’ is derived from the neuter nominative of
the verbal adjective ‘Xextoc’, which in turn is derived from the verb ‘X6yetv’, to
say or to speak. Since lektos is one of those adjectives having the sense both of
the perfect passive participle and of the notion of possibility, to lekton is probably
best understood either as ‘that which is said’ or as ‘that which can be said’. Now
it is true that one way in which the Stoics characterise the lekton is to say that it
is ‘that which is signified’ (to arjpaivovevov) by a significant utterance (e.g. AM
8.11-12), and some writers take it that to lekton ought therefore to be rendered,
either exclusively or primarily, as ‘what is meant’. 95 This interpretation, which
gains further credence from the fact that Liddell and Scott list ‘to mean’ as one
of the senses of legein, is then taken to imply that a lekton was thought of by the
Stoics as some kind of ‘meaning’ or ‘sense’, whatever such may be (e.g., Graeser
[1978b]).
95 Andreas Graeser, for instance, asserts that “in Stoic semantics [the verb semainein] stands
exclusively for a relation that holds between the linguistic sign and its sense” [Graeser, 1978b,
p. 8l]. Since he also takes semainomenon to be synonymous with lekton [Graeser, 1978b, p.
87], it is apparent that he would give preference to this reading. The Kneales, who argue that
the Stoics “deliberately identified semainomena with lekta," are of the opinion that “‘what is
meant’ is probably the most literal translation of lekton ” [Kneale and Kneale, 1962a, p. 140].
According to Bocheiiski, “the Xcxtov corresponded to the intension or connotation of the words”
[Bocheriski, 1963, p. 84].
The Megarians and the Stoics
459
There are at least two difficulties with this reading. One is the question whether
the Stoics did in fact take ‘crrifioavopevov’ and ‘Xexxov’ to be synonymous. From
the fact that they called lekta semainomena , it does not follow that all significata
of significant utterances are lekta. Indeed, there is a passage in Diogenes Laertius
(7.58) which seems to make it clear that this is so. According to this passage, what
is signified by a name is an ‘individual quality’ (18ia tioi6tt]c;), and by a common
noun, a ‘common quality’ (xoivf) jimoxrjc). Now since the qualities of corporeal
bodies are, according to the Stoics, as much material as the bodies themselves (cf.
Simplicius in cat. 217.32), and since lekta are not material entities, what seems to
be suggested is that ‘what is signified’ by a name or common noun is not a lekton.
In other words, ‘to arjpouvopsvov’ is not coextensive with ‘Xexxov’.
The other problem with reading to lekton as ‘what is meant’ is that there is more
than a little evidence supporting the idea that one species of lekton, the axidma,
had the role in Stoic semantic theory as that which is true or false. Hence, if one
interprets the lekton as being in general a ‘meaning’ or ‘sense’, then one seems to
commit the Stoics to saying that such things as ‘meanings’ or ‘senses’ are the sorts
of things which can be true or false. We think that one would be hard pressed to
find textual evidence for such a commitment. Thus it would appear that however
else they may have thought of the axidma, it is unlikely that the Stoics could
have viewed it as such a thing as a meaning or sense. It seems to us rather that
the Stoics would have agreed with Austin in his contention that “we never say
‘The meaning (or sense) of this sentence (or of these words) is true’” (“Truth” in
Phil. Papers, 87); hence, it seems unlikely that the Stoics could have viewed the
axidma as such a thing as a meaning or sense. But if it is improbable that the
Stoics thought of the axidma as a meaning or sense, then since the axidma is a
kind of lekton, it is not clear that one can legitimately promote an interpretation
of the lekton as being in general a meaning or sense. From the point of view of
the interpretation we are suggesting, rendering ‘to lekton ’ as a meaning or sense
cannot do justice to the various roles the concept plays in Stoic semantics (cf.
[Long, 1971, p. 77]).
But even supposing we interpret l to lekton ’ as ‘what is said’, there is still some
controversy whether we should also interpret it as ‘what can be said’. The problem
is summarised by Andreas Graeser as follows:
For taking lekton to mean “that which can be said” may seem tan¬
tamount to committing oneself to the position that the lekton exists
regardless of whether it is being expressed or not, whereas taking lekton
to mean “what is said” seems rather to entail that the very entity in
question exists only as long as the expression that asserts it [Graeser,
1978b, pp. 87-88],
This worry is reiterated later in Graeser’s essay when he asks “Did the Stoics
hold that the axiomata or lekta respectively exist in some sense whether we think
of them or not?” [Graeser, 1978b, p. 95], and it is echoed by A.A. Long when
he wonders whether “ lekta only persist as long as the sentences which express
460
Robert R. O’Toole and Raymond E. Jennings
them” [Long, 1971, p. 97]. In giving expression to this problem both Graeser and
Long are concerned to reply to an assertion made by the Kneales to the effect
that axiomata “exist in some sense whether we think of them or not” [Kneale and
Kneale, 1962a, p. 156]. The context in which this claim is made is an iteration
of the various similarities and dissimilarities perceived by the Kneales between
axiomata and propositions. We shall not comment on the arguments adduced
by the Kneales concerning this issue, nor on the counter-arguments presented by
Graeser and Long. Indeed, we intend to develop an interpretation of the lekton
which will require an understanding of the meaning of the term l to lekton ’ as being
systematically ambiguous. On such a reading this controversy would seem to be of
less concern. We do, however, wish to note that saying “lekta only persist as long
as the sentences which express them” does not seem to render their existence any
less mysterious or problematic than saying that they “exist in some sense whether
we think of them or not.”
The passage in Diogenes Laertius at 7.66, discussed in the last section, would
seem to suggest that the axidma is the significatum of some actual utterance of a
particular type, i.e., an assertion. Similarly, each of the other kinds of lekta is the
significatum of the appropriate type of utterance (i.e., command, question, and
so on). It seems apparent that on this account the question whether axiomata
“exist in some sense whether we think of them or not” should not arise, for the
subsistence of the axidma is clearly dependent upon the existence of an act of
assertion. Obviously this dependent status will apply to the lekta corresponding
to the various other types of illocutionary acts. We suggested in the last section
that one ought to understand ‘ lekton ’ in these contexts as a generic term denoting
the content of a speech act; hence, it seems appropriate in such cases to take ‘to
lekton ’ to mean ‘what is said’.
On the other hand, there would seem to be room in the Stoic theory for lekta
which subsist independently of any particular utterance. When one asserts, for
example, that the state of affairs {Dion, walking} is a fact, or commands that
it become a fact, or questions whether it is a fact and so on, what gets said,
or exhibited (jiapacnfjaou) in language, in such an utterance is the unarticulated
objective content (to cpavxoccrdcv) of the rational presentation, that is, the Tipaypa
{Dion, walking} (cf. AM 8.70). We have proposed that 1 lekton’ is sometimes used
to denote the pragma which is the unarticulated content of a rational presentation.
In such contexts, it seems appropriate to understand ‘to lekton’’ to mean ‘what can
be said’.
The lekton and ontology
There is a tradition among ancient commentators that the Stoics posited a sum-
mum genus (yevixotoitov) which they called ‘the something’ (to ti) (AM 8.32; PH
2.86; Seneca epist. 58.13-15), and under which they included not only material
bodies or ‘corporeals’ (acoporra), but also a set of items ‘without body’ which they
The Megarians and the Stoics
461
called ‘incorporeals’ (aacofiaiot) (AM 10.218). 96 We are informed by Sextus Em¬
piricus that under the class of incorporeals were included lekta, void, place, and
time (xevov, xottov, xpbvov) (AM 10.218). 97 Now inasmuch as the Stoics thought
that “bodies alone are existents,” 98 it is apparent that they did not take ‘to be
something’ necessarily to mean the same as ‘to exist’ in the sense that material
bodies exist.
In addition to material bodies and incorporeals. the class of somethings appears
to have included a collection of items containing both fictional beings and theo¬
retical constructs, particulars such as Centaurs (Seneca epist. 58.15) and limits
(Proclus SVF 2.488; DL 7.135). Although it might seem natural to assume that
these particulars ought to have been classified among the incorporeals, there is no
evidence to support the view that the Stoics did so, for none of the texts providing
a list of the incorporeals include such items in the list. The fact that the members
of this class of ‘mental constructs’ are included among the ‘somethings’ but are
never included among the incorporeals, would seem to indicate that the genus of
‘the something’ was differentiated into three subclasses: the class of material bod¬
ies or ‘corporeals’; the class of ‘incorporeals’ which included lekta, void, place, and
time; and the class of fictions or mental constructs (cf. Long and Sedley [1990,
1.163-66]). At least one respect in which such a tripartite differentiation would
be significant is to give lie to the claim made by some modern commentators that
the incorporeals were viewed by the Stoics as merely ‘constructs of the mind’ (e.g.,
Watson, [1966, pp. 38-39]). We believe that the commentary of Long and Sedley
is sufficient to show that the Stoics did indeed propose this tripartite division of
the genus of ‘the something’. But granting this division as a component of Stoic
ontology, the question occurs as to the basis for differentiating between fictional or
theoretical constructs and the incorporeals. Since an adequate treatment of this
problem is beyond the scope of this essay, we can only give a suggestion here of
the reason.
It seems fairly clear, at least with respect to void, place, and time, that the
Stoics needed these items in their ontology in order to develop their physical
and cosmological theories. A consideration of the roles envisaged for these items
makes it also seem clear that although these incorporeals fell short of having the
real existence that substantial bodies have, it is unlikely that the Stoics viewed
them merely as mental constructs. Similar reasons can be adduced on behalf of
the lekton, supposing that Frede is correct in his suggestion that the concept of the
lekton was first introduced in the ontology of Stoic causal theory [Frede, 1987a,
p. 137]. On the other hand, the connection between the lekton and the immanent
96 For additional citations from the primary sources, see SVF 2.329-35; AM 10.234; AM 11.224;
PH 2.223-25; Plutarch adv. colot. 1116b-c. For commentary on these notions by modern writers,
see Long and Sedley [1990, pp. 162-66]; Long [1971, pp. 88-90]; Rist [1969a, pp. 152-54]; Watson
[1966, pp. 92-96]; Sandbach [1975, p. 92],
97 cf. Plutarch, adv. colot. 1116B.
98 ovxa yap pova xa aopaxa xaXouoiv(Plutarch comm. not. 1073e). cf. Aetius plac. 1.11.4,
4.20 (SVF 2.340, 387); Stobaeus eclog. 1.336, 338; Cicero acad. 1.39; Seneca epist. 117.2, 106.4;
DL 7.56.
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Robert R. O’Toole and Raymond E. Jennings
logos , a feature of its role as the pragma which is the content of a presentation,
would seem to provide further reason why it is unlikely that the lekton was viewed
merely as a mental construct.
Recalling our suggestion that for the Stoics ‘to be something’ did not seem
to mean the same as ‘to exist’, the question naturally arises as to what ‘to be
something’ did mean. It has been suggested that for the Stoics ‘to be something’
meant “to be a proper subject of thought and discourse” (Long and Sedley [1990,
1.164]). This idea is developed with the observation that since the Stoics thought
that expressions such as ‘Centaur’ and ‘limit’ “are genuinely significant, they are
taken to name something, even though that something has no actual or indepen¬
dent existence” [Long and Sedley, 1990, 1.164]. It is not obvious, however, what
the force of the expression “genuinely significant” is supposed to be in this con¬
text. This shortcoming, however, can probably be filled out by a consideration
of what is excluded from the genus of the something. We have a passage from
Stobaeus (eclog. 1.136.21 = SVF 1.65) which would seem to indicate that the
Stoics did not include what ‘the ancients’ (oi dp^aiot) called ‘ideas’ (iSecti) in the
class of somethings. Michael Frede plausibly suggests that these ‘ideas’ which
the Stoics called ‘concepts’ (evvoripara), are the transcendental Ideas or Forms of
Plato ([Frede, 1987b], 348). According to Stobaeus, ‘concepts’ such as ‘Man’ or
‘Horse’ were referred to by Zeno and his followers as ‘pseudo-somethings’ (cboavef
TLvot). A possible reason why these items might have been refused the ontological
status of somethings can be gleaned from a passage of Sextus Empiricus. Clearly
presenting Stoic doctrine, Sextus argues at AM 7.246 that the genera of which
the particular instances may be of this kind or that kind cannot themselves be
of either kind. Thus the generic ‘Man’ is neither Greek nor Barbarian, for if he
were Greek, then all particular men would have been Greek, and, conversely, if he
were Barbarian, then all particular men would have been Barbarian. We take the
general point of this argument to be the idea that it is not possible to ascribe to
the ennoemata any of the attributes one may ascribe to the particulars which fall
under them. But if one cannot say of the universal ‘Man’ that he is either Greek
or Barbarian, young or old, tall or short, cowardly or brave, and so on, then the
term ‘man’ would seem to lack ‘genuine significance’ when it is used in this way.
Hence the force of the expression “genuinely significant” might be understood to
specify a contrast between terms such as ‘Centaur’ and ‘limit’, which are taken
to name items to which one can ascribe certain appropriate attributes, and terms
such as ‘Man’, which are taken to name items to which one can not ascribe such
attributes. Thus, although it makes sense to say ‘A Centaur has four legs’, it does
not make sense to say ‘“Man” has two legs’. Hence, the Stoics might have thought
that an item such as a Centaur or a limit could be said to be something, which
is to say “the proper subject of thought and discourse,” but it was evidently not
part of their ontological commitment to think that an item such as the universal
‘Man’ could also be so."
"Note that the Stoic use of the expression ‘universal Man’ is as a synonym for the expression
’the concept “Man”’. We should remind ourselves that such items belong to an ontological
The Megarians and the Stoics
463
The terms which the Stoics standardly used in their characterisations of the
incorporeals were various forms of the verbs ‘OcpLaxaaDoa’ (AM 8.70; DL 7.63), 100
and ‘xaputpujxattyfloti’ (AM 8.12; Simplicius in cat. 361.10). These terms, which are
both customarily translated as ‘to subsist’ (e.g., Long and Sedley [1990], 1.196,
162-66), are contrasted with the verb ‘uxotpyeLv’ (e.g., Stobaeus eclog. 1.106.20)
which, on at least one of its senses, can be translated as ‘to exist’. This distinction,
referred to by Galen as ‘splitting hairs’ (SVF 2.322), was, needless to say, the
source of much critical commentary (cf. also Alexander in topica 301.19). We shall
not attempt here to discuss this criticism, since it has, in any case, already been
adequately addressed by A.A. Long [1971, pp. 84-90]. It seems clear, however, that
the Stoics used this distinction to indicate the ontological status of the incorporeals
as ‘somethings’, although not necessarily as existents. It has been suggested that
the Stoic usage of the terms “seems to capture the mode of being that Meinong
called bestehen and Russell rendered by ‘subsist’” (Long and Sedley [1990, 1.165]).
The parallel is perhaps even closer inasmuch as the Stoics also seemed to count
‘fictions’ or ‘mental constructs’ such as surfaces and limits as belonging to the
class of ‘somethings’ and to use forms of these verbs to refer to them (Proclus SVF
2.488; DL 7.135). It would be wrong, though, to infer from this that they classed
incorporeals as fictions, the views of some modern commentators notwithstanding.
6 AXIOM AT A
It is evident that in some respects axiomata have a character similar to that of
propositions. For one thing, several texts confirm the judgment that the Stoics at¬
tributed to axiomata the property of being true or false. 101 There is some question,
however, whether axiomata were true or false in ‘the basic sense’. This question
arises because the Stoics assigned the terms ‘true’ and ‘false’ not only to axiomata ,
but also to arguments (Xoyoi) and to presentations (cpavxocafat). An argument was
said to be true whenever it was conclusive (auvaxxixov) and had true premisses
(PH 2.138; DL 7.79), and, according to Sextus, “a true presentation is one from
which it is possible to produce a true predication (xaxqyopia), 102 such as this in
the present circumstances: ‘It is day’, or this: ‘It is light’” (AM 7.244). 103 The
category different from that to which the common quality ‘Man’ belongs, the latter, according
to the Stoics, being something corporeal.
100 cf. Cleomedes SVF 2.541; Proclus SVF 2.521; Stobaeus eclog. 1.106.19.
101 For example, Sextus Empiricus: AM 8.10; 12; 73; 74; Diogenes Laertius: 7.65; 68.
102 Martha Kneale observes that “if we take xaTYjyopta here as equivalent to alpcopa then Sextus
is defining the truth of presentations in terms of the truth of axiomata ” [Kneale and Kneale,
1962a, p. 150]. Although admitting that “this identification is plausible” [Kneale and Kneale,
1962a, p. 150], she is hesitant however to apply it, her reason being that the term ‘kategoria'
appears only this once in Sextus’ writings, and its meaning is nowhere mentioned by him [Kneale
and Kneale, 1962a, p. 150]. But it seems that many commentators think that this identification
is more than plausible (e.g., Mates [1953, p. 34]; Long and Sedley [1990, 1.240]; Long [1971, p.
92]; Graeser [1978a, p. 201]).
103 &Xrp9eu; pcv ouv daiv tov ecttiv aXr)0fj xaT7)yopiav Ttot^atxaOai, <1 k tou ‘fipcptx eotiv’ tm tou
xapovxoc r] tou ‘cpolc ecttiv.’
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Robert R. O’Toole and Raymond E. Jennings
consensus among modern commentators, however, seems to be that “the basic ap¬
plication was probably to propositions” [Long, 1971, p. 92], In addition to being
the primary items to which the terms ‘true’ and ‘false’ are applied, axiomata are
like propositions in that they are signified by declarative sentences. 104 According
to the Kneales, two further ways in which axiomata are similar to propositions,
is that “they are abstract, or, as the Stoics perhaps rather unhappily put it, in¬
corporeal; and they exist in some sense whether we think of them or not” [Kneale
and Kneale, 1962a, p. 156]. We have discussed the latter thesis in Subsection 5.5.
As for the former, it may be that we can plausibly think of axiomata as ‘abstract’;
it would seem, however, that we can criticise the Stoics for calling them ‘incorpo¬
real’ instead of ‘abstract’ only if we are certain that they meant ‘abstract’ and not
‘incorporeal’.
At any rate, however many of these characteristics of propositions one wants to
apply to axiomata, there are several differences which, according to the Kneales,
indicate that axiomata cannot simply be identified with propositions [Kneale and
Kneale, 1962a, pp. 153-56]. For one thing, axiomata appear to have certain ‘gram¬
matical’ characteristics which we usually do not associate with propositions, but
rather with the sentences which express them. For another thing, lekta obviously
have moods. For another, lekta in general, and hence axiomata in particular, have
tenses. This is indicated by the titles of a series of four books written by Chrysippus
and reported by Diogenes Laertius. These titles are Temporal Lekta 105 , too (3ooxc
(ITepi xov xorca xpovoup \eyo\iivov a' P') and Axiomata in the Perfect Tense , two
books (Ikpi ctuvtsXixov d^topaxov p') (DL 7.190). There are reports as well that
predicates, which are major constituents of axiomata, were distinguished according
to voice (DL 7.64-65) and number (Chrysippus SVF 2:99.38-100.1). Michael Frede,
in his discussion of the origins of traditional grammar, has suggested that for the
Stoics the notion of syntax was applied primarily to lekta and only derivatively to
parts of speech and sentences ([Frede, 1987b], 345-47). Hence, distinctions which
we would expect to be made at the level of expressions are made by the Stoics
at the level of lekta, and the features at the level of expressions which correspond
to certain features distinguished at the level of lekta, take their names from these
latter features ([Frede, 1987b], 345). If Frede is correct, then it should come as no
surprise that axiomata differ from propositions in these ways.
Another difference between axiomata and propositions, is that axiomata can
change truth value. As the Kneales point out, this feature is what might be
expected since axiomata have tenses [Kneale and Kneale, 1962a, pp. 153-54].
Finally, axiomata may cease to exist. 106 The evidence for this latter property
104 The examples of axiomata which Sextus cites at AM 8.93-98 and which Diogenes Laertius
cites at 7.68-70 are all clearly signified by declarative sentences. The texts at AM 8.71 and DL
65-66 also indicate that axiomata are signified by declarative sentences.
105 Here we are translating ‘to XEybpevov’, which literally means ‘that which is spoken’ as
‘ lekton A.A. Long has said that “in sense lekton can hardly be distinguished from to XcyopEvov”
([Long, 1971], 107nl3).
106 cf. Graeser’s remarks on these differences between axiomata and propositions [Graeser,
1978b, pp. 94-95],
The Megarians and the Stoics
465
is a passage of Alexander of Aphrodisias (in an. pr. 177.25-178.1). He reports
that according to Chrysippus, the axioma ‘This man is dead’ (indicating Dion
demonstratively) is impossible when Dion is alive but is ‘destroyed’ (cpDdpea'flcd)
when Dion has died (177.31). On the other hand, the axioma ‘Dion has died’
which is possible when Dion is alive is apparently still possible when Dion has
died (178.21-22). This result is what one would expect, given that a demonstra¬
tive must indicate the individually qualified substrate (ISlox; jioiov ), that is, the
qualified substance of Dion, whereas the name signifies the individuating quality
(fSiot Ttoioxrjc) (DL 7.58). When Dion has died, the individually qualified substrate
ceases to exist as such, and thus can no longer be indicated demonstratively; the
individuating quality, however, can still be referred to by the name.
According to Sextus Empiricus, the ‘dialecticians’ (i.e., the Stoics) declare that
the first and most important distinction among axiomata is that between those
which are ‘simple’ (cntXa) and those which are ‘complex’ (ou^ omXa) (AM 8.93).
Sextus reports that even though axiomata are constructed of other elements, they
are called ‘simple’ if they do not have axiomata as constituents. Thus a simple
axioma is one which is neither constructed from a single axioma taken twice (81c;
XapPavopsvov), nor from different axiomata by means of one or more conjunctions
(auvSeapop) (AM 8.94). The following, for example, are simple axiomata, as is
every axioma of similar form: ‘It is day’, ‘It is night’, ‘Socrates is conversing’
(AM 8.93). Complex axiomata , on the other hand, are those constructed from a
single axioma taken twice, for example, ‘If it is day, it is day’; or those constructed
from different axiomata by means of a conjunction, for example, ‘If it night, it is
dark’. Further examples of complex axiomata are such as the following: ‘Both it
is day and it is light’, ‘Either it is day or it is night’ (AM 8.95). The content of
these passages should be compared with the similar content of the text of Diogenes
Laertius at 7.68-69. Each author goes on to discuss the various kinds of simple
and complex axiomata, but we shall refer to Diogenes’ text for this information.
Simple axiomata, according to Diogenes, are classified as follows: ‘negative’ (to
dmocporuxdv) , ‘negatively assertoric’ (to apvqxixov), ‘privative’ (to oTepiycixov), ‘as-
sertoric’ (to xorcr]Y 0 P tx ° v ) > ‘demonstrative’ (to xaTccyopeuTixov), and ‘indefinite’
(to aopiaxov) (7.69). 10 ' In the passage at 7.70, Diogenes provides some details
about these various kinds of simple axiomata. A negative axioma is constructed
with a negative particle and an axioma , e.g., ‘Not: it is day’. A negatively asser¬
toric axioma is produced from a negative constituent and a predicate, e.g., ‘No
one is walking’. A privative axioma is constructed with a privative constituent
and a possible axioma (odjuapcaoc xara 8uvapiv), e.g., ‘This man is unkind’. An
assertoric axioma is constituted by a nominative case and a predicate, e.g., ‘Dion
is walking’. A demonstrative axioma is constructed with a demonstrative nom¬
inative case and a predicate, e.g., ‘This man is walking’. An indefinite axioma
consists of one or more indefinite constituents and a predicate, e.g., ‘Someone
is walking’. Diogenes makes a special note of the ‘double-negative’ axioma (to
UTCEpaTroqxxTixov) . This is a negative axioma constructed with a negative particle
107 We have followed the translations of Long and Sedley ([1990], 1.205) to render these terms.
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Robert R. O’Toole and Raymond E. Jennings
and a negative axioma, e.g., ‘Not: not: it is day’. Such an axioma, according to
Diogenes, is assumed to have the same meaning as the axioma ‘It is day’ (DL
7.69).
Of the complex axiomata described by Diogenes, we will consider only those
which have a role in the Stoic syllogistic system. 108 These are the ‘conditional
axioma’(t‘o ouvqppevov), the ‘disjunctive axioma’ (to SieCeuypevov), and the ‘con¬
junctive axioma' (to aujiTtETtXeypivov). The conditional axioma is constructed
by means of the conditional connective ‘if’ (ei). 109 This connective ‘guaran¬
tees’ (EiayyeXkTai) that the second constituent of the conditional axioma follows
(axoXoufteiv) from the first, as, for example, ‘If it is day, it is light’ (DL 7.71).
A disjunctive axioma is constructed by means of the connective ‘or’ (t)tol). This
connective guarantees that one or the other of the constituent axiomata is false,
for example, ‘Either it is day or it is night’ (DL 7.72). A conjunctive axioma is
constructed by means of a ‘conjunctive’ connective, such as the particle ‘xai’ in
this example: ‘It is day and it is light’ (xai qpepa ecra xai cpaic; ecru) (DL 7.72).
A topic of interest at this point might be that concerning the truth conditions
for the various types of complex axiomata. We shall discuss the truth conditions
for the conditional in the section on inference, but for the moment we intend to
consider some questions about axiomata in relation to what has been written about
presentations and pragmata. One of the distinctions among presentations recorded
by Diogenes Laertius is that between sensory (aich)r|TLxat) and non-sensory (oux
aicrdrpxat) presentations (DL 7.51). He writes that “sensory presentations are
those apprehended through one or more sense-organs, whereas non-sensory pre¬
sentations are those perceived through the mind itself, such as those of the incor-
poreals and of other things apprehended by reason” (DL 7.51). 110 In Passage A
Sextus implies that a Greek speaker, on hearing a significant utterance in Greek,
will apprehend the pragma signified and subsisting coordinately with thought,
whereas the non-Greek-speaker will not apprehend the pragma (AM 8.12). We
interpreted this as a description of how a rational presentation would be induced
in the mind of the Greek speaker by the utterance. The content of the presentation
would be the pragma signified by the utterance.
We take it that such a presentation, although induced by a sound sensed through
the hearing organs, or perhaps by the marks on a papyrus or a stone sensed through
the organ of sight, would, nevertheless, be classified as a non-sensory presentation.
For in order that a presentation be a sensory presentation, it seems evident that not
only must it be apprehended through one or more of the sense organs, and hence
have its cause in some portion of the qualified substrate, but also it must have
108 Sextus Empiricus’ account of complex axiomata is not so compact or concise as that of
Diogenes, but it is perhaps more philosophically interesting. He talks about the conditional
axioma at AM 8.108-12, and about the conjunctive axioma at AM 8.124-29, but he does not
seem to have an account of the disjunctive axioma which is comparable to that of Diogenes.
109 This doxography is attested by Chrysippus in his Dialectics and by Diogenes the Babylonian
in his Art of Dialectic (DL 7.71).
110 aiaOiyuxc(i prev a! 8i’ aiaDrjxripiou fj aitxflTjxrjpiuiv XapPavopEvat, oux aiat)r|xixai 8’ ori Sia xfjc
Siavoiac xaDaxep xGv aatopaxiov xai x£>v aXXwv xOv Xoyoi Xapflavopivov.
The Megarians and the Stoics
467
a content in which either that portion of the qualified substrate is itself signified
by a demonstrative, or the quality which individuates it is signified by a name.
It is apparent that in the normal course of events, a presentation caused by an
utterance may satisfy the first requirement, but it will not satisfy the second, for
the content of such a presentation, as is indicated by the discussion in the preceding
paragraph, is the pragma signified by the utterance, and not any feature of the
utterance itself.
There is evidence that the Stoics viewed certain thought processes as some
sort of ‘internal discourse’ (EvStorflexoc Xoyix). 111 Such thoughts can no doubt
be considered as ‘utterances’, and as such, will induce presentations in the mind.
Clearly the presentations produced by such utterances will be non-sensory. Hence
it would seem that a non-sensory presentation may be induced in one’s mind either
by someone (else) speaking a pragma , or by one speaking a pragma in thought.
The pragma spoken (icpotypa Xextov) might be an axidma signified by an assertion,
but it might also be prostaktikon signified by a command, or an erdtema signified
by a query, or some other type of lekton. No doubt we not only sometimes make
assertions to ourselves in thought, but also sometimes ask ourselves questions or
exhort ourselves to action. This latter notion of speaking imperatives to ourselves
in thought seems to be a necessary feature of Brad Inwood’s interpretation of the
Stoic theory of action [inwood, 1985, pp. 59-60 86-87], Interesting as it might be
to follow up on these other classes of lekta , axiomata, however, are the lekta which
are of interest in the present context.
We take it that the presentation induced by someone uttering a declarative
sentence, or by someone uttering a declarative sentence in thought, will have a
pragma as content which has a structure isomorphic to that of the axidma signified.
We are using ‘isomorphic’ here to suggest a structure preserving correspondence
between the elements of the pragma and the elements of the axidma. Thus, if we
see Dion walking and so have a sensory presentation of Dion walking, the pragma
accompanying this presentation can be represented by the simple structure {Dion,
to walk}; however, if someone were to say to us ‘Dion is walking’ so that we have
a non-sensory presentation of Dion walking at the present time, the attendant
pragma , although representable on some level by the same structure, would seem
to require a representation which includes an element to signify the present tense.
111 In the Theatetus Socrates says that “when the mind is thinking, it is simply talking to
itself, asking questions and answering them, and saying yes or no” (190a). In a similar vein in
the Sophist, the Eliatic Stranger says that “thinking and discourse are the same thing, except
that what we call thinking is precisely, the inward dialogue carried on by the mind with itself
without spoken sound” (263a). No doubt there are problems with the view that all thinking
is like internal discourse; it seems, however, that something of this tradition was carried on by
the Stoics, for according to Sextus Empiricus, they held that “it is not with respect to uttered
speech (;ipo<popix<x Xoyoc) that man differs from the irrational animals (for crows and jays
and parrots utter articulate sounds), but with respect to internal discourse (EvStordeTOC Xoyoc)”
(AM 8.275). And according to Galen, the Stoics define the mental process which provides the
means of converting sensory data to knowledge, that is, the process by means of which “we
understand consequence and conflict, in which separation, synthesis, analysis and demonstration
are involved” (SVF 2.135), as ‘internal discourse’ (endiathelos logos).
468
Robert R. O’Toole and Raymond E. Jennings
One might, for example, portray this pragma as follows: {Dion, to walk: now}.
However, since it is beyond the scope of this work to develop a detailed account
of how such representations might be handled, we will resist the temptation. We
have chosen to characterise the axioma by merely enclosing its signifying sentence
in braces, and although there are probably good reasons to develop distinct modes
of representation for the pragma and the axioma, we have chosen to depict the
pragma by the same method.
To consider a more complex example, suppose that someone says to one ‘Dion
was walking about in Athens at noon yesterday’. It seems evident that the pragma
which accompanies the presentation induced by this utterance will reflect the struc¬
ture of the axioma , and hence have as constituents not only the individuating
quality signified by the name ‘Dion’ and the predicate signified by the verb ‘walk¬
ing’, but also temporal components and the individuating quality signified by the
name ‘Athens’. We shall represent both the pragma and the axioma as follows:
{Dion was walking about in Athens at noon yesterday}. It is worth emphasising
that we shall represent the pragma in this way only when it is the content of a
presentation induced by an utterance.
7 THE CONDITIONAL AXIOMA
The conditional axioma , according to Chrysippus and Diogenes the Babylonian, is
constructed from two axiomata by means of the connective ‘if’ (el) (DL 7.71). Of
the two constituent axiomata , the one signified by the sentence placed immediately
after the connective is called the ‘antecedent’ and ‘first’ (f]youpevov), whereas the
other is called the ‘consequent’ and ‘second’ (XfjY ov ) (AM 8.110). 112 The connec¬
tive ‘if’ seems to ‘promise’ or ‘guarantee’ (CTaYYEXXeXetctt) that the consequent
‘follows’ (axoXouflei) the antecedent (AM 8.111; DL 7.71); hence, the relationship
of ‘following’ (dxoXouflla) between its antecedent and consequent is evidently the
characteristic property of the conditional axioma. Since the conditional axioma
is, after all, an axioma, it might be expected that for any particular conditional
axioma, one could give an account of the presentation to which it corresponds,
and of the pragma which is the content of the presentation. A passage of Sextus
Empiricus, which we intend to quote presently, provides a clue to the psychologi¬
cal aspects of this relationship. However, since the context in which this passage
occurs is a discussion of the Stoic doctrine of signs, it might be useful to give a
short summary of this teaching.
According to Sextus Empiricus, the Stoics define the sign (to oqpeiov) as a true
antecedent axioma in a sound conditional, capable of revealing (exxaXuitTixoc;)
the consequent (AM 8.245; 250; PH 2.104). 113 Sextus reports that signs were
distinguished between those which are ‘indicative’ (evSetxxixov), and those which
112 Sextus notes that reversing the normal sentence order does not affect this rule. Thus, in
each of the examples ‘If it is day, it is light’ and ‘It is light, if it is day’, the antecedent is the
axioma signified by the sentence ‘It is day’ (AM 8.110).
113 At AM 8.104, Sextus implies that this definition was reserved only for the ‘indicative sign’
The Megarians and the Stoics
469
are ‘commemorative’ (uTiopvrjcrtixov) (AM 8.151). An indicative sign is said to
indicate ‘that which is naturally non-evident’ (to (puact aSrjXov), and is never
observed in conjunction with the thing signified (8.154). The soul, for example,
is naturally non-evident, and its existence is supposed to be indicated by bodily
motions (8.155). A commemorative sign, on the other hand, signifies what is
‘temporarily non-evident’ (ext iov Ttpoc xatpov a8f]Xcov), and is sometimes observed
in conjunction with what is signified; hence, the perception of the sign brings to
mind what is often perceived along with it but is momentarily unperceived. For
example, since smoke is often observed in conjunction with fire, it is taken as a
commemorative sign of fire even though the fire itself is unperceived (8.151-52).
7.1 Akolouthia: psychological aspects
The connection between the doctrine of signs and the notion of ‘following’
(dxoXoufKot) is spelled out by Sextus Empiricus in a passage which records the
Stoic reply to several criticisms levelled at the theory of signs by the Skeptics.
[The Stoics] say that it is not with respect to uttered speech (xpo-
cpopixo<; Xoyoc;) that man differs from the irrational animals—for crows
and jays and parrots utter articulate sounds, but with respect to in¬
ternal discourse (evStdfiexoc Xoyoc). Nor [does man differ from the
irrational animals] with respect to the simple presentations (for they
also form such presentations), but with respect to the ‘inferential’
(peiaPcaixri) 114 av8 'copitoaixtovaX' (auvfieiixrj) presentations, because
of which he immediately possesses the conception (evvoia) of ‘following’
(dxoXoufita), and through the conception of following he apprehends
the notion (v6r]atc) of sign (arftieiov); for sign itself is such as this: ‘If
this, then this’. Therefore it follows that sign also exists in accordance
with the nature and constitution of man (AM 8.275-77).
We interpret this passage as follows. The faculty of forming presentations from
our conceptions and complex presentations from simpler ones is part of the nature
and constitution of human beings. This faculty is itself founded on our capacity
for ‘internal discourse’, which makes possible the ‘inferential’ and ‘compositional’
thought necessary for the production of such presentations. Thus, we differ from
the irrational animals, for they do not possess this faculty for producing presenta¬
tions, but must rely solely on their senses for the data from which presentations
are formed. Moreover, because of this capacity for constructing inferential and
compositional presentations, we also possess the conception of ‘following’, and
since the relationship between the sign and what it reveals can be represented as a
conditional axidma, it is through the conception of following, that we understand
the notion of sign.
114 The adjective ‘pexaPocuxoc’ is derived from the noun ‘pcxdpaau:’, the basic meaning of which
is ‘a moving over’ or ‘a shifting’ or ‘a change of position’ (Liddell and Scott). This etymology is
reflected in the use of the adjective in this context which seems to suggest the transition from
one conception to another by the process of inferential or discursive thought
470
Robert R. O’Toole and Raymond E. Jennings
Of immediate interest in this text is the statement that we possess the con¬
ception of akolouthia because of our ability to form inferential and compositional
presentations. There is little indication in the passage as to how this capacity is
supposed to produce the conception of following. It may be possible, however, to
work out an interpretation by considering some clear examples of akolouthia from
Sextus Empiricus’ discussion of the theory of signs, and by keeping in mind the
quotation of Aetius concerning the acquisition of conceptions and preconceptions,
as well as the texts of Diogenes Laertius and Sextus Empiricus concerning the
ways of producing complex conceptions. 110 Our conjecture is that the ‘inferential’
and ‘compositional’ presentations are those from which the conceptions of general
conditionals are inscribed on the soul. Furthermore, we would suggest that it is
the totality of conceptions of general conditionals from which the conception of
following arises.
Consider, then, some examples of signs mentioned by Sextus Empiricus. The
following are commemorative signs: smoke is a sign of fire and a scar a sign
of a previous wound (AM 8.152-53; PH 2.102). A punctured heart is a sign of
immanent death (AM 8.153). Lactation is a sign of conception (AM 8.252; PH
2.106) and a bronchial discharge is a sign of a lung wound (AM 8.252-53). In
these examples, according to Sextus, the sign often appears together with what
it indicates, and hence, when the latter is not evident, the sign is able to reveal
it because we remember that they occur together, for instance, that a punctured
heart results in death (AM 8.152-53). But it would seem to be implicit in the
passage quoted above that a certain degree of prior conceptual development must
take place before one acquires the conception of sign, and so understands one state
of affairs as a sign of another states of affairs.
This development, as Sextus indicates, no doubt begins by one’s noticing that
certain types of presentations seem to occur together as a sequence. Thus a pre¬
sentation of a man who has been wounded in the heart will be followed after some
period of time by a presentation of the same man having died. The pragmata
which are the contents of these presentations may be represented respectively by
the complex {This man, to be wounded in the heart} and the complex {This man,
to die}. They would be spoken respectively as the axiomata This man is wounded
in the heart and This man has died (cf. AM 8.254). Given some number of
similar situations and supposing the capacity for ‘inferential’ and compositional’
thought—perhaps along with certain conceptions and preconceptions already es¬
tablished, for example, general conceptions of causality—one might produce a non-
sensory presentation of a causal connection between these types of events. 116 The
content of this presentation would be a complex pragma , and the components of
this complex pragma may be represented as follows: the first constituent will be
115 See Section 4.2 for both of these references
116 We are making two assumptions here which we have not made explicit but which seem
plausible. These are (1) that a basic notion of causality would be recognised by the Stoics as a
preconception (upoXrjilnc) acquired by most people, and (2) that the Stoics would recognise the
examples of commemorative signs listed above as cases in which there is a causal connection
between the sign and what it indicates
The Megarians and the Stoics
471
the pragma {Someone, to be wounded in the heart}, followed by, say, an arrow to
represent the connection, 117 followed by the pragma {That one, to die}. Finally,
the whole complex will be enclosed in braces. Thus the representation will be
constructed as follows:
{{Someone, to be wounded in the heart} —» {That one, to die}}.
The axioma which is this pragma spoken as an assertion might be represented in
a similar manner, although it may contain certain constituents such as temporal
elements not present in the representation of the pragma. The axioma, then, might
be represented as follows:
{{Someone is wounded in the heart} —> {That one will die}}.
Taking the particle ‘if’ (ei) as the connective which seems to provide the most
natural way to signify the arrow, the sentence which signifies this axioma will
be the sentence ‘If someone is wounded in the heart, that one will die’. This
sentence which signifies a generalised conditional axioma (to xocdoXixov) might
be seen as expressing a law-like relationship between those situations in which
someone is wounded in the heart and those situations in which that person dies
of the wound. No doubt this could be viewed as a relationship of ‘following’ or
‘consequence’ in the causal sequence of events, and given a similar analysis of
the causal relationships between other states of affairs, it seems probable that a
general conception of following would be developed; moreover, one might plausibly
assume that the relationship of following conceived to obtain between events in
the causal nexus would be carried over to the axiomata. In any case, there is no
doubt that the Stoics viewed the relationship of akolouthia as one which holds
between axiomata as well as between events in the causal nexus.
A difficulty with this interpretation is that for most of the examples cited from
Sextus Empiricus, the direction of the relationship between the sign and what it
indicates, or between the antecedent and consequent of the parallel conditional
axioma, is not the same as the direction of the causal sequence. For example, a
scar is said to be a sign of a previous wound. The general conditional axioma
might be expressed by the sentence ‘If anyone has a scar, that one has had a
wound’. According to the interpretation so far, having a wound would follow from
having a scar, but clearly, the direction of akolouthia with respect to the causal
sequence is from the occurrence of the wound to the formation of the scar. So it is
obvious that some adjustment must be made in this account of the development
of the conception of akolouthia.
Previously, we quoted A.A. Long ([1971, p. 46]: see page 425) to the effect
that since the Stoics assumed that events occurred according to a strict causal
nexus, they perhaps assigned to logic as its major function the task of making
possible predictions about the future from considerations of what follows from
117 The arrow will turn out to be the item at the level of lekta which is signified by the connective
‘if’ (ei)
472
Robert R. O’Toole and Raymond E. Jennings
present events. On this assumption, one might expect that they would have con¬
centrated at first on examples in which the direction of akolouthia coincided for
the causal sequence and the relationship of following in the conditional axioma,
and so stressed the development of the conception of following as we have inter¬
preted it above. But of course, they would also have been interested in drawing
out the present consequences of past actions or events, especially with respect to
allocating responsibility in the sphere of ethics. Hence, the following relationship
between the antecedent and consequent of a conditional axioma need not always
proceed from cause to effect as does the following relationship of the parallel causal
sequence. And not the least consideration would be those instances of following
between the parts of a conditional axioma which do not correspond to instances
of following in the causal nexus: in other words, conditional axiomata matching
logical connections. Nevertheless, it is not implausible to suppose that the con¬
ception of following as it relates to axiomata had its basis in a preoccupation with
the kinds of examples which involve reasoning from cause to effect.
There are not so many examples of indicative signs occurring in the text as
there are of commemorative signs, but here are two: bodily movement is a sign of
the presence of the soul (AM 8.155) and sweat flowing through the skin the sign
of the existence of intelligible pores (AM 8.306). For the Stoics, the first example
would be a straightforward instance of a causal relationship. The presence of the
cause, however, must be inferred from the existence of the effect, hence acquiring
the conception of the relationship between them will depend entirely on already
established conceptions and preconceptions, and on the capacity for inferential and
compositional thought. The relationship in the second example is not obviously a
causal relationship, but seems to be a strictly logical. Yet acquiring the conception
of the following relationship between these states of affairs would also seem to
depend on previously acquired conceptions and preconceptions, such as theories
about surfaces and the flow of fluids, as well as on inferential and compositional
thought.
7.2 Pragmata spoken as conditionals
Suppose that someone utters the sentence ‘If Dion is walking, Dion is moving’. 118
What is the nature of the pragma signified by this utterance and of the presentation
which has this pragma as content? Apparently, the utterance of this sentence will
induce a presentation in the mind of the hearer, the content of which will be a
complex pragma. The components of this complex pragma may be represented
as follows: the first constituent will be the pragma {Dion, to walk}, followed by,
say, an arrow to represent the relation of akolouthia, 119 followed by the pragma
{Dion, to move}. Finally, the whole complex will be enclosed in braces. Thus the
representation will be constructed as follows:
118 cf. DL 7.78 where this conditional is featured as the major premiss of the argument ‘If Dion
is walking, Dion is moving; but Dion is walking; therefore, Dion is moving’
n9 The arrow is the item at the level of lekta which is signified by the connective ‘if’ (d)
The Megarians and the Stoics
473
{{Dion, to walk} —» {Dion, to move}}
As we indicated in the previous section, the axioma, that is, the pragma spoken,
will be represented by the construction
{{Dion is walking} —» {Dion is moving}}
The information conveyed by the axioma and apprehended in the pragma is that
any situation in which Dion is walking will be one in which Dion is moving; more¬
over, this information is communicated regardless of whether there ever is any real
situation in which Dion is walking.
Similarly, the utterance of the sentence ‘If Dion was walking about in Athens at
noon yesterday, he was not on the Isthmus at noon yesterday’ will induce a presen¬
tation which has as content the pragma which may be represented by the complex:
{{Dion was walking about in Athens at noon yesterday} —»
{Dion was not on the Isthmus at noon yesterday}}.
What the axioma conveys is that the existence of the situation in which Dion
was walking about in Athens at noon yesterday, rules out the existence of the
situation in which Dion was on the Isthmus at noon yesterday. 120 We shall see
in the sequel that the standard criterion for the sound conditional axioma , that
is, one for which the consequent follows from the antecedent, is a reflection of the
kind of relationship that holds between these situations. This relationship might
also have been signified by a sentence constructed with the connective ‘not both
... and ... but it may be that Chrysippus wished to reserve this connective to
signify contingent relationships between states of affairs. 121
Each of the conditional axiomata in these examples will be sound (uytec) or
true (&Aryd£<;) if and only if the consequent axioma follows from the antecedent
axioma, which is to say, if and only if it is the case that the pragma signified by the
consequent sentence follows from the pragma signified by the antecedent sentence.
How would someone know that this relation holds? In the first example, if one were
to have the conception of the general conditional axioma If anything is walking,
that thing is moving, and were to know that the relation holds for the general
conditional, then it seems clear that one would know that it holds for the particular
conditional. It is probably important that this conception could, in principle, be
acquired by experience, although it is probably more likely that it would be taught.
It could be acquired by experience because in Stoicism something walking possesses
an attribute which is identifiable as a certain configuration or mixture of the
120 The Stoics characterise the relationship between these situations by saying that they ‘conflict’
(pdxEicu) with one another
121 cf. Mueller [1978, p. 20]; Kneale and Kneale [1962a, p. 16l]; Long and Sedley [1990, 1.211].
Note that the relationship between these situations could not be signified by the Stoic disjunction,
since that connective indicates that exactly one of the disjuncts is true. But evidently it need
not be the case that Dion be either at Athens or on the Isthmus.
474
Robert R. O’Toole and Raymond E. Jennings
pneuma with the individually qualified substrate; moreover, something moving also
has an attribute identifiable in the same manner. Thus one could find out through
direct experience that anything which possessed one attribute also possessed the
other, and in this way come to have the conception of the general conditional. In
the other example, one would rely on the conception of the general conditional
{{Something is at place A at time t} —> {that thing cannot be at place B at time
t}}.
It is unclear how the acquisition of this conception would be explained. It may
be that it would be classified as a preconception (rrpoXrppic)- In any event, knowing
that the relation holds for the particular conditional would seem to require knowing
that it holds for the general conditional.
8 SEMANTICS AND INFERENCE
The foundation of the Stoic system of inference was the so-called ‘indemonstra-
bles’ (ot avootoSEixxoi), a set of five basic argument types which are attributed
to Chrysippus by several sources (AM 8.223; DL 7.79; Galen inst. log. 34-
35), although there is some controversy whether they actually originated with
him. Their origin has been ascribed to the Peripatetic philosopher Theophrastus
by both Prantl and Zeller, 122 but it has been argued by Bochenski 123 and oth¬
ers 124 that this claim is doubtful. On the other hand, there is some indication
that arguments of this sort were discussed by Aristotle and his followers. In the
Prior Analytics at 50 a 16-50 6 4, Aristotle discusses arguments from agreement
(e£ opoXoYiac) and arguments by reduction to the impossible (Sid xqc Eic aSuvcrrov
dmaywYfjc). 125 He says that many other arguments are concluded from hypothesis,
and these he promises to consider and distinguish in the sequel. But, as Alexander
of Aphrodisias points out (in an. pr. 390.1), Aristotle never did fulfil this promise.
Alexander conjectures, however, that perhaps Aristotle was speaking of certain
arguments from hypothesis mentioned by Theophrastus and Eudemus and some
others of his followers (390.2-3). These include ‘arguments by connection’ (xouc
xe 8id ouv£$ouc), which are also called ‘conditional’ (auvqppEvov) or ‘hypothetical
by an additional premiss’ (xouc; Sid xrjc TtpoaXrjijiEGx; utcoOexixouc), ‘arguments by
122 Zeller [1962, 119n2]; Mates [1953, p. 69] attests that both Prantl and Zeller held this view,
and Kieffer [1964, p. 66] confirms that Prantl held it. He cites Volume 1, page 386 of Prantl’s
Geschichte der Logik im Abendlande (Graz, 1955). (Photographic reprint of Leipzig, 1855,
edition).
123 Both Mates [1953, p. 69] and Kieffer [1964, p. 8] cite Bochenski’s La Logique de Thu)opraste
(Collectanea Friburgensia, N.S. fasc. xxxvii) (Fribourg: 1947). Mates cites pages 116-17 and
chapter 7, note 5; while Kieffer cites page 103 ft
124 Kieffer writes that Prantl “by a strained interpretation of certain passages in the later com¬
mentators on Aristotle .. . reached the unjustified conclusion that the traditional hypothetical
syllogisms were the discovery or invention of Theoprastus” ([Kieffer, 1964], 66). It seems clear
that Mates ([1953], 69) also rejects Prantl’s conclusion.
125 This is the wording of Alexander of Aphrodisias (in an. pr. 389.31).
The Megarians and the Stoics
475
separation’ (too 8ia tou StatpETLXou), also called ‘disjunctive’ (SieCeuypevov), and
‘arguments from a negated conjunction’ (roue Sia dmocpcmxfj? aupxXoxfjc) (390.3-
6). In another place, Alexander attributes this alternative terminology to ‘the
younger philosophers’ or neoteroi (vEcAcepot), and there is no doubt that this is
Stoic terminology. The terms ‘auvqppEvov’, ‘StE^Euypcvov’, and ‘aupxXoxf) axo-
cpotTixrj’ are attested as Stoic in numerous places. The term ‘xpoiuxov’ is attested
by Sextus Empiricus (PH 2.202) and by Galen ( inst. log. 17.1), and the term
‘xpoaXf)t]x<;’ by Sextus (AM 8.413) and by Diogenes Laertius (7.76). According to
KiefFer [1964, p. 66], the terms ‘argument by connection’ (Sict tou auveyouc;) and
‘argument by separation’ (8ta tou StaipETixou) are not found in Aristotle’s works.
KiefFer [1964, p. 67] cites Bochenski’s argument (La Logique de Thu)ophraste 108)
that since these terms are Peripatetic but not Aristotelian, they likely were coined
by Theophrastus and Eudemus. All in all it would seem to be rather unclear to
what extent these philosophers advanced the study of hypothetical arguments. If
indeed they were responsible for this terminology, then perhaps, as the Kneales
suggest [Kneale and Kneale, 1962a, p. 105], its existence is an indication that
they made some headway in the analysis of such arguments. On the other hand,
we have the testimony of Boethius that neither Theophrastus nor Eudemus car¬
ried the investigation into the hypothetical syllogisms much further than where
Aristotle left off (Graeser Die logischen Fragmente des Theophrast, fr. 29).
A general description of the indemonstrables would be that they are argu¬
ments with two premisses of which the ‘major’, called ‘ tropikon ’ (xpoiuxov), is
either a conditional, a disjunction, or a negated conjunction, and the minor, called
‘proslepsis ’ (xpoaXr^u;), is a categorical. If the major is a conditional, then the
minor is either its antecedent, in which case the conclusion is its consequent, or
it is the negation of the consequent, in which case the conclusion is the negation
of the antecedent. If the major is a disjunction, then either the minor is one of
the disjuncts, in which case the conclusion is the negation of the other disjunct,
or it is the negation of one of the disjuncts, in which case the conclusion is the
other disjunct itself. If the major is a negated conjunction, then the minor is one
of the conjuncts and the conclusion is the negation of the other conjunct. The
indemonstrables were often represented by the Stoics as they are below, that is,
as schemata having ordinal numbers as variables: 126
126 See Mates [1953, p. 68] (Table II) for an extensive documentation of the evidence for the
indemonstrables. Strictly speaking, I should be talking about the ‘conditional axioma’, the
‘disjunctive axioma’, and the ‘conjunctive axioma’, but for the most part we shall refer to
‘conditionals’, ‘disjunctions,’ and ‘conjunctions’ in the sequel.
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Robert R. O’Toole and Raymond E. Jennings
(I)
If the first, the second;
(IV) Either the first or the second;
but the first;
but the first;
therefore, the second.
therefore, not the second.
(II)
If the first, the second;
(V) Either the first or the second;
but not the second;
but not the first;
therefore, not the first.
therefore, the second.
(III) Not both the first and the second;
but the first;
therefore, not the second.
In what follows we intend to consider mainly those arguments having a conditional
axioma as the major premiss, its antecedent as minor premiss, and its consequent
as conclusion, which is to say, those arguments exemplifying schema (I) in the
above list.
8.1 The sound conditional
In this section we consider several topics concerning the character of the Stoic
conditional. These questions would seem to have a bearing on one’s view of the
role of the conditional in the Stoic system of inference and on one’s view of Stoic
logic in general. These topics are: (1) the debate over the criterion for a sound
conditional, (2) the question whether there was a standard criterion for the Old
Stoa, and (3) an account of the notion of conflict (jadtxT]) which appears in the
criterion attributed to Chrysippus.
The controversy
It is well known to students of ancient logic that in the fourth century B.C. a con¬
troversy prevailed among various ‘dialecticians’ 127 as to the proper criterion for a
sound conditional (to uyi£<; auvrjjipevov ) 128 Although this dispute is mentioned
briefly by Cicero in Academica 2.143, our information comes mainly from the writ¬
ings of Sextus Empiricus. At 8.108 in Adversus Mathematicos, Sextus outlines the
Stoic characterisation of the conditional axioma . 129 The conditional, according
127 This reference to the ‘dialecticians’ would seem to be one in which the term should be taken
in the general sense of ‘one who practices dialectic’ or ‘logician’, rather than in the sense of
denoting a member of the Dialectical School (cf. footnote 20, page 408).
128 In his translation of Sextus Empiricus, R.G. Bury renders ‘auvr)(rpevov’ as ‘hypothetical
syllogism’; however, as Mates points out ([Mates, 1953], 43), this term always denotes an ‘if . .. ,
then ... ’ proposition in the examples given by ancient commentators, never an argument or
inference-schema.
The Stoics seem to use ‘uyiec’ (sound) and ‘dXrjflsc’ (true) interchangeably in these contexts.
We shall argue in the sequel that we should understand ‘true’ in the sense of ‘sound’ when it so
occurs, rather than the other way around.
129 The Greek is ‘alfiiona’. This term is often translated as ‘proposition’ but we have not com¬
mitted ourselves to so translate it (see page 423). It would seem better for us to transliterate
The Megarians and the Stoics
477
to the Stoics, is a non-simple axidma whose parts are joined by the connective
‘if’ (ei). The part preceding this connective is called the ‘antecedent’ or ‘first’
(rpfoupevov), whereas the other is called ‘consequent’ or ‘second’ (Xrjyov). Such an
axidma “seems to promise that its second component follows consequently on its
first” (AM 8.111), and if this promise is carried through so that the consequent
does indeed ‘follow’ the antecedent, then the conditional is true (diXrplec ), but if
not, then it is false ((jteuSot;) (AM 8.112).
Note that in these passages where he is discussing the controversy on condition¬
als (AM 8.112-17; PH 2.110-12), Sextus appears to use ‘uyte<;’ (sound) and ‘dXrjfis?’
(true) as though they were synonymous. In the glossary of Stoic Logic, Mates
claims that, according to Stoic usage, these terms are interchangeable ([Mates,
1953], 132). Jonathan Barnes states that the conditional “‘If p then q' is uyufy; iff
it is true.” On the other hand, he also writes that “of course, ‘uyif|<;’ does not mean
‘true’” ([Barnes, 1980], 169nll). It seems possible that there is an ambiguity in
Sextus’ use of the adjective ‘otXryMt;’, which, given that he is reporting on Stoic
doctrine, one might assume to be a reflection of an ambiguity in Stoic usage. Sex¬
tus seems for the most part to represent the Stoics as using this term to describe
a statement expressing an axidma the content of which correctly represents ‘the
way things are’. For example, he reports in one place that the definite axidma
‘This man is sitting’ or ‘This man is walking’ is true (dXryOec;) whenever the person
indicated by the demonstrative is actually sitting or walking (8.100). 130 On the
other hand, there is some evidence to suggest that in the passages cited above
(AM 8.112-17; PH 2.110-12), he may also use the term (dXr]'0£<;) in the sense of
‘genuine’ or ‘real’.
First, there is the evident fact that he seems to use ‘oyii<;’ and ‘dXryOec’ inter¬
changeably. But, as Barnes has pointed out, ‘uyt£<;’ does not mean ‘true’ in the
sense of a correct representation of the ‘way things are’. On the other hand, ac¬
cording to other sources on Stoic logical theory, it does appear in certain contexts
to mean ‘genuine’ or ‘proper’. Consider, for example, the following dilemma set
out by Aulus Gellius (2.7.6-10):
A father’s command is either honourable or base;
if his command is honourable, it is not to be obeyed merely because it
is his order, but because it is right that it be done;
if his command is base, it is not to be obeyed because what is wrong
ought not to be done;
therefore, a father’s command ought never to be obeyed.
Gellius rejects this argument on the basis that the leading premiss “cannot be
considered what the Greeks call a sound and proper (uyiec; et vopipov) disjunctive
proposition.” (2.7.21). He claims that it requires the additional disjunct “or are
neither honourable nor base” in order to be considered a genuine Stoic disjunction.
the term, since at least part of our thesis involves the question of its meaning.
130 Similarly, Diogenes Laertius relates that on the Stoics account, the axidma ‘It is day’ is true
(&Xr|-0£c) just in case it really is day, otherwise, it is false (i^cOSoc) (DL 7.65).
478
Robert R. O’Toole and Raymond E. Jennings
Gellius’ motivation here seems to be a reflection of his claim at 16.8.12-14 that the
disjuncts of a Stoic disjunction must exhaust the alternatives.
Sextus himself makes the same claim at AM 8.434 where he includes an argu¬
ment invalid because of deficiency in his classification of invalid Stoic arguments. 131
Here Sextus writes that in the following argument the disjunctive premiss is defi¬
cient:
Either wealth is an evil or wealth is good;
but wealth is not an evil;
therefore wealth is good.
In order to be uyi.ec; , according to Sextus, the disjunctive premiss ought to read
as follows: ‘Wealth is either a good or an evil or indifferent’. In both this example
and in Gellius’ example, the etymological derivation from the sense of ‘Oytec;’ as
‘sound’ to its sense as ‘complete’ or ‘having the required characteristics’ would
seem to be clear. Moreover, since that which is incomplete could not be a proper
exemplar of its kind, the derivation to its sense as ‘genuine’ also seems clear.
At AM 8.111-112, Sextus writes that a conditional proposition “seems to promise”
that its consequent follows (axoXoufleT) from its antecedent. Moreover, he adds
that such a proposition is aXrjflec; just in case this promise is fulfilled. Since a con¬
ditional proposition which lacked this characteristic could not be a proper Stoic
conditional, one might suppose that a conditional is also therefore Oytec; when this
promise is fulfilled. Now inasmuch as ‘aXrjflec’ can be used in the sense of ‘genuine’
or ‘real’, it is possible, therefore, that Sextus uses both ‘Oytec’ and ‘otXrydec’ in that
sense. This would explain why he uses them interchangeably, even though ‘uyt£c’
does not mean ‘true’ in the sense of a correct representation. However, a con¬
ditional proposition which was ‘sound’ or ‘true’ in the sense of ‘genuine’—which
is to say that its consequent follows from its antecedent—could not help but be
‘true’ in the derivative sense that it would correctly represent the real connections
between the states of affairs represented in the antecedent and consequent.
To return to Sextus’ account, it would seem to be agreed among the dialecticians
that a conditional axidma is sound whenever the consequent ‘follows’ from the
antecedent. The disagreement arises, however, over the question of a criterion to
determine when this relation of following holds. At AM 8.112, Sextus sets up the
dispute as follows:
Now on the one hand all the dialecticians assert in common that the
conditional proposition is sound whenever its consequent follows its
antecedent. On the other hand, concerning when and how it follows
they are at odds with one another, and set forth conflicting criteria for
the notion of ‘following’. 132
131 According to the classification of invalid arguments reported by Sextus at PH 2.146 and AM
8.429, Gellius’ argument in its unaltered form is also an example of an argument invalid because
of deficiency.
132 xoivffic [rev y“P cpacuv ocxavxec; ol StaXexxtxol uyisc eTvat auvr)[i[ievov oxoev axoXot/drj xS ev ocuxfi
t)You[ievm xo ev auxO XfjYOvr))= xepi 8e xou koxe axoXouOei xai jiffic axaaia^oucu itpoc aXXr|Xou<;.
xai [layopEva xfjc axoXou'fliac exxiflevxai xprnqpia.
The Megarians and the Stoics
479
The most valuable discussion of the controversy over the criterion for a sound
conditional is presented by Sextus in the passage at PH 2.110-12, wherein he
outlines the four distinct and competing accounts. In this presentation Sextus
apparently orders these definitions from the weakest to the strongest, in each case
citing an example which is allowed by the next weaker interpretation, but which
is rejected by the one under discussion.
Sextus begins by summarising the position of Diodorus Cronus of the Dialec¬
tical School and the conflicting position of his pupil, Philo the Dialectician. 133
He attributes the first account to Philo, and states that according to this ver¬
sion, a conditional is sound whenever it is not the case that the antecedent is
true and the consequent false (PH 2.110, cf. AM 8.113). In the passage at AM
8.113, Sextus presents what is in effect a truth table for the Philonian conditional.
According to this summary, there are three combinations of truth values for the
components of the conditional which make it sound and one which makes it false.
These assignments correspond to the assignments in the truth table for the mate¬
rial conditional (cf. AM 8.245); consequently, there is general agreement among
modern logicians that Philo’s definition amounts to a definition of the material
conditional (cf. Mates [1953, p. 44]; Bochehski [Bochenski, 1963, p. 89]). The
second definition cited by Sextus is ascribed to the Diodorus Cronus. According
to Diodorus, a sound conditional is one which neither was capable nor is capable
of having a true antecedent and a false consequent (PH 2.110; cf. AM 8.115).
Mates has argued cogently that a sound Diodorian conditional is an always true
Philonian conditional ([Mates, 1953], 44-46).
Sextus attributes the third version of the correct criterion to those who advance
the view that there must be a ‘connexion’ or ‘coherence’ (auvapTrjau;) between
the antecedent and consequent of a sound conditional. According to this view,
a conditional proposition is sound whenever the contradictory (dvuxetpevov) 134
ocp ltc covaexuevx covqAi^xc; (pot)(T)Tai) with its antecedent (PH 2.111). Unlike the
first two cases, this definition is not linked by Sextus to the name of any particular
philosopher. Recently, however, several authors (e.g., Kneale and Kneale [1962a],
129; Gould [1970], 76; Mueller [1978], 20) have cited a passage in Cicero [De Fato ,
12) as evidence that the ‘connection’ view is that of Chrysippus. We shall refer to
this definition of a sound conditional as the ‘connexivist view’, in accordance with
its attribution by Sextus to “those who introduce connexion.” 135
The fourth definition, according to Sextus, is advocated by “those who intro¬
duce ‘implication’ (epcpdau;)■” It states that in a sound conditional the consequent
133 In the passage AM 8.112-17 Sextus gives a more detailed account of the differences between
Diodorus and Philo, but he does not include any mention of the other competing views.
134 In his glossary, Mates points out the distinction between ‘dvxixdpevov’, which means the
contradictory of a proposition, and ‘oiTCOipaxixov’, which means a proposition with ‘not’ prefixed
to it. His example makes the distinction clear: “The propositions ripepa eaxi'v [It is day] and oux
fipepa eaxtv [It is not day] are both avxtxdpcvov with respect to one another, but only the latter
is anocpaxixov’ ([Maxec, 1953], 133).
135 The name has been adopted in certain modern interpretations such as those of Storrs McCall
(e.g., in Anderson and Belnap [cl975-, pp. 434—52] and in McCall [1966],
480
Robert R. O’Toole and Raymond E. Jennings
must be ‘potentially contained’ (7t£pi£)(£Tat Suvapci) in the antecedent (PH 2.112).
According to Mates ([1953], 49), this fourth definition cited by Sextus is not dis¬
cussed by any other ancient sources, nor has its ancestry been attributed to any
particular philosopher. In addition, with such a dearth of information, it has been
little discussed by modern commentators. Martha Kneale has suggested that this
may even be a Peripatetic view (Kneale and Kneale [1962a], 29). Long and Sedley,
on the other hand, think that it may not be significantly different from the con¬
nexion account ([Long and Sedley, 1990], 1.211). In any event, this version does
not bear the name of any ancient philosopher, nor has a name been adopted as a
consequence of its modern interpretations, as in the case of the connexivist thesis.
More recently, however, Michael J. White has speculated that the motivation for
this fourth type of conditional is somewhat akin to the ideas put forward in mod¬
ern relevant logics (White [1986, pp. 9-14]; hence we might call this fourth view
the ‘relevantist’ view. However, since White’s speculations are somewhat tenuous,
and since the philosophers who propose the definition invoke the notion of the
virtual ‘inclusion’ or ‘containment’ of the consequent in the antecedent, it would
seem better to call this fourth view the ‘inclusion’ or ‘containment’ criterion.
As an example of a conditional which is sound according to Philo’s criterion,
Sextus cites the following: ‘If it is day, I converse’. This conditional is sound, he
says, when in fact it is day and the subject is conversing (PH 2.110). And indeed, if
Philo’s definition is the analogue of the material conditional, it would also be sound
whenever either it is not day or the subject is conversing. But Sextus tells us that
according to Diodorus this conditional is false (epeuSoc;), 136 since it is obviously
capable of having a true antecedent and false consequent whenever it is in fact
day, but the subject remains silent (PH 2.110). As is the case with each of the
critiques offered by Sextus, one has to consider the possibility that this objection
was not in fact put forward by Diodorus but was contrived by Sextus himself for
exegetical reasons. It was noted above that in presenting these definitions of a
sound conditional Sextus’ intention seems to have been to order them from the
weakest to the strongest, one definition being stronger than another just in case
an example can be found which is rejected as being a sound by the former, but
which is accepted by the latter. Martha Kneale [1962a, p. 129]) has pointed out
that if Sextus did so arrange them, then it cannot be assumed that these criteria
were actually conceived in the order presented. But even if one cannot make this
assumption, it seems to us that one can put forward an account of the development
of the controversy which is at least partially along the lines of Sextus’ arrangement.
For one thing, it is unclear why chronological priority should be a factor in the
debate between Diodorus and Philo. Since these philosophers were teacher and
pupil, then regardless of which definition was put forward first, it seems plausible
136 In his discussions of conditionals, Sextus seems for the most part to use ‘t^EuSoc’ to contrast
with both ‘uyiEc’ and ‘aArjDec’ (cf. AM 8.112-17 and PH 2.110-12). On the other hand, he also
uses ‘pox^rjEoc’ for this purpose (cf. PH 2.105, 111), but this latter term seems to be used more
extensively to mean ‘invalid’ or ‘faulty’ in connection with arguments (PH 2.150; AM 8.433) or
argument schemata (PH 2.146, 147; AM 8.429, 432).
The Megarians and the Stoics
481
to suppose that it was Diodorus himself who articulated the objections to Philo’s
account and put forward the counter-example. In addition, since several modern
commentators agree that the connexion view can be attributed to Chrysippus, it
seems feasible that this criterion was formulated later than both the Diodorian
and the Philonian definitions; moreover, it seems quite reasonable to suppose that
it was Chrysippus who raised the objections to the Diodorian view. On the other
hand, it would be somewhat more difficult to substantiate Sextus’ ordering of the
connexion and inclusion accounts, the reason being that there is no confirmation
other than Sextus’ own testimony to support the hypothesis that the inclusion
criterion was formulated after the connexion account. Nor is there any other
evidence to support his version of the inclusionist objections to the connexivist
criterion. Hence, in contrast to Kneale’s assumption that “we can take it that the
objections mentioned by Sextus were in fact put forward at some time” [Kneale
and Kneale, 1962a, p. 129], we would urge that one not take his account of the
debate between the inclusionists and the connexivists for granted.
The conditional presented by Sextus as being sound according to Diodorus’
criterion but not sound according to the connexion criterion is the following: ‘If
it is not the case that atomic elements of existents are without parts, then atomic
elements of existents are without parts’ 137 (nH 2.111). Tr)t<; a^iopa coouXS pe
xpue ov AtoSopuo' accouvT Pccauae it aXwaijic (otei) 138 begins with the false clause
‘It is not the case that atomic elements of existents are without parts’ and ends
with the true clause ‘atomic elements of existents are without parts’; hence, it
never was capable, nor is it capable of beginning with a true antecedent and
ending with a false consequent (PH 2.111). It seems clear that the axioma ‘Atomic
elements of existents are without parts’ is conceptually or analytically true, and
hence necessary. What is more relevant, however, is that it would count as a
necessary proposition according to the versions of necessity of both Diodorus and
Chrysippus. The definition of Diodorus is worded as follows: “The necessary is
that which being true, will not be false” ( necessarium, quod cum verum sit non
1 37 » 5 V 5 />, -V, V r* V 5 M V ~
El oux ecttiv ocpEpr) tuv ovtiov aiotxeia, eanv apepf) tcov ovtcovotoixeici.
Bury translates this as: ‘If atomic elements of things do not exist, atomic elements exist’,
whereas Martha Hurst has a reading similar to ours [Hurst, 1935, p. 489]. Mates appears to
agree with Bury’s translation, and his argument for this reading is as follows: “[W]e are explicitly
told that the denial of the consequent is not incompatible with the antecedent. Since the denial of
the consequent is the antecedent, this implies that the antecedent is not incompatible with itself.
But if the antecedent were the negation of an analytic statement, it would be incompatible with
itself’ [Mates, 1953, p. 50]. The problem with this argument is that Mates is assuming a ‘non-
connexivist’ interpretation of ‘incompatible’. According to this view, any necessary proposition
is incompatible with itself. But this is just the assumption that the connexivists wish to deny
(cf. page 482).
It is perhaps worth mentioning that Hurst cites this example as evidence in her argument
against a temporal reading of Diodorus’ definition of a sound conditional [Hurst, 1935, p. 489],
the temporal interpretation being the one favoured by Mates. In doing so, however, we believe
she errs in not taking seriously the possibility that this is a counter-example brought against the
Diodorean criterion by the connexivists, and not necessarily an example which Diodorus would
have put forward himself.
138Bury fails to translate the Greek word for ‘always’, thereby missing the point of the example.
482
Robert R. O’Toole and Raymond E. Jennings
erit falsum ) (Boethius in de interp. 234); whereas that of Chrysippus is worded
thus: “The necessary is that which being true does not admit of being false, or
admits of being false but is prevented by external factors from being false” (DL
7.75). 139 It is evident that on either account of necessity, Diodorus’ criterion
for a sound conditional will make the counter-example sound merely by the fact
that the consequent is necessary or that the antecedent is impossible, since either
circumstance is sufficient to insure that the conditional never was capable, nor
is capable, of having a true antecedent and a false consequent. Thus it seems
plain why Diodorus’ definition would be rejected by someone who thinks that a
sound conditional requires a connexion or coherence between the antecedent and
consequent, for clearly his criterion would permit a conditional to be sound even
though there is no connection whatever between its parts.
Note that the rejection of the counter-example cited in the previous section can
be generalised by stating that the connexivist criterion renders false any condi¬
tional in which the antecedent and consequent are contradictories. This charac¬
teristic property of the connexivist view of implication is stated by Storrs McCall
as follows: “[N]o proposition connexively implies or is implied by its own nega¬
tion, since it is never incompatible with its own double negation, nor is its own
negation incompatible with itself’ [McCall, 1966, p. 415]. According to McCall,
“this connexive property of propositions was known to Aristotle” [McCall, 1966,
p. 415]. In the Prior Analytics Aristotle argues that “it is impossible that the
same thing should be necessitated by the being and by the not-being of the same
thing” (57 63 ). If it is supposed, for example, that if A is white, then necessarily B
is great, and if A is not white, then necessarily B is great, then “it follows of neces¬
sity that if B is not great, then B itself is great; but this is impossible” (aupPcdvei
dvayxrjc too B |iEydXou Hfi ovxoc auxo xo B elvai qeya tou S’ aSuvaxov) (57 H3 ).
Consequently, McCall dubs this property, which he represents in Polish notation
as NCNpp , ‘Aristotle’s Thesis’ ([McCall, 1966], 415). 140
139 dvayxaTov 8e ecjxiv oitep aXflec ov oux ccmv eniSexxixov toO (JjeOSoc eTvai, fj £7 ci5exxix6v pev
ectti, xa 8’ exxoc auxco Evavxiouxai xpoc xo <|>e08o<; eTvcu. This account of the necessary is not
specifically attributed to Chrysippus by Diogenes; however, as Mates points out, a passage of
Plutarch (de Stoic repugn. 1055d-e) would seem to indicate that this view cited by Diogenes is
that of Chrysippus.
Compare the view called ‘Stoic’ by Boethius (in de interp. 235): necessarium autem, quod
cum verum sit falsam praedicationem nulla ratione suscipiat (The necessary is that which when
it is true, by no account will admit of a false affirmation). According to Martha Kneale [1962a,
p 123], this version “can safely be attributed to Chrysippus” since the context in which it occurs
in Boethius is similar to that in which Cicero (de fato 12-20) contrasts the views on modality of
Chrysippus and Diodorus.
140 Another characteristic connexive property mentioned by McCall is ‘Boethius’ Thesis’, repre¬
sented as: CCqrNCqNr ([McCall, 1966], 416). In De syllogismo hypothetico, Boethius presents
an inference schema which McCall takes to require the assumption of the connexivist principle
for a sound conditional. The schema is this: If p, then if q then r; if q then not-r; therefore,
not-p [McCall, 1966, p. 415]. If we take Cpq and CpNq as connexivist conditionals, then p is in
conflict with both Nq and NNq. Hence, it seems reasonable on connexivist grounds to say that
Cpq and CpNq are in conflict, and by the connexivist definition for a sound conditional, we get
CCqrNCqNr, i.e., Boethius’ Thesis. Presumably, then, Boethius’ argument proceeded as follows
The Megarians and the Stoics
483
As an instance of a conditional sound according to the connexivist criterion,
Sextus puts forward the example ‘If it is day, it is day’ (PH 2.111). This conditional
is connexively sound since obviously every proposition must be in conflict with its
own contradictory. Sextus claims that on the inclusion view this proposition and
every such ‘duplicated’ (Siatpopooficvov) conditional would be false, the reason
being that “it is not feasible that any object should be included in itself’ (PH
2.112). Sextus does not give an example of a conditional sound according to
the inclusion criterion, and the reason may be, as is suggested by Michael J.
White, that “it would ill accord with his purpose of producing suspension of belief
... with respect to all accounts to leave the reader with the impression that
[this] last account ... is correct” [White, 1986, p. 10]. White also suggests that
Sextus “gives the impression of having to strain a bit” [White, 1986, p. 10] in
his attempt to show that the aforementioned connexivist paradigm would be false
according to the inclusion view. As he expresses the point, this rather “literal¬
minded” interpretation of ‘ii£pie)( eTQ(t Suvotpei’ does nothing to convince one that
these conditionals were indeed rejected by the inclusion view. 141
It is of interest in this regard to note some comments which have been recorded
concerning the relationship between the connexion and inclusion conditionals. The
remark of Long and Sedley to the effect that they may not differ significantly from
one another has already been mentioned above. Add to this the comments of
Martha Kneale that “the difference between them was small” and that “the ob¬
jection which the partisans of implication brought against the theory of connexion
is not of a fundamental kind” (Kneale and Kneale [1962a, p. 134]), as well as the
observation of Mates that “the fourth type of implication seems to be a restricted
type of Chrysippean implication” [Mates, 1953, p. 49], and there seems to be
reason enough to concur with White’s doubts about the accuracy of Sextus’ re¬
port concerning the relationship between these two accounts of implication. Given
these doubts, one is tempted to speculate that the order of appearance between
the connexion and inclusion definitions may have been the reverse of Sextus’ ar¬
rangement. If so, then it may be that Chi vsippus saw a need for more precision
than that afforded by the inclusion definition, and thus formulated the connexion
account in response to that perception.
(cf. McCall [1966, p. 416]):
(1)
P -> (q -> r)
Assumption
(2)
q —»• -ir
Assumption
(3)
(q —> r) —»■ -(q ->• ->r)
Boethius
(4)
-■-’(q -> -’«•)
2 Double Negation
(5)
'(u -> r )
3,4 Modus Tollens
(6)
_, p
1,5 Modus Tollens
141
Commenting on Sextus’ statement that the inclusion definition
the form ‘If p, then p’, Michael Frede says that “Sextus Empiricus himself makes it very clear
his comment is merely his own interpretation of the definition” [Frede, 1974, p. 90].
484
Robert R. O’Toole and Raymond E. Jennings
The standard Stoic conditional
There are several reasons to suppose that from Chrysippus on, the connexivist
account was the standard doctrine of the Old Stoa concerning the criterion for
a sound conditional. First, there is some textual evidence in Diogenes Laertius.
At 7.71, in recording the Stoic account of non-simple axidmata, Diogenes reports
that according to Chrysippus in his Dialectics and Diogenes of Babylon in his
Art of Dialectic , a conditional is a non-simple axioma constructed by means of
the connective ‘if’ (el), such that this connective promises that the consequent
follows (axoXoudcTv) the antecedent. Later, at 7.73, he attests that according to
the Stoic criterion, a true (aXryflEc) conditional is one in which the contradictory of
the consequent conflicts (pa^CTotu) with the antecedent. Now although this version
of a sound conditional is not actually attributed to Chrysippus and Diogenes of
Babylon, the juxtaposition of these passages would seem to indicate a connection.
In any case, there is further indirect textual support afforded by the passage of
Cicero at De fato 12. We have already mentioned that this passage has been cited
by several modern commentators as providing strong evidence that the connection
or coherence criterion was the account accepted by Chrysippus. 142
Although the testimony of Cicero and Diogenes Laertius would appear to be
sufficient to establish that Chrysippus propounded the connection doctrine for a
sound implication, it does not focus any light on the question as to what crite¬
rion Zeno and Kleanthes supported. Indeed, the information available in the texts
would seem to be inadequate to establish any certainty in this regard. However,
there are some passages in which Sextus Empiricus attributes the Philonian crite¬
rion to ‘the Stoics’ or to ‘the Dogmatists’, and these remarks are taken by at least
one writer to indicate that Zeno adopted Philo’s definition of a sound conditional
[Rist, 1978, p. 391]. Since Zeno and Philo were contemporaries and both students
of Diodorus Cronus, this is a plausible conjecture. On the other hand, there is
nothing specific in these passages to link the Philonian criterion to Zeno, and in
fact, there is room for some doubt concerning Sextus’ attribution of this definition
to the Stoics.
As a sceptic Sextus was out to discredit the views of all the so-called ‘dogmatic
philosophers’. With respect to the Stoics this would have involved, among other
things, showing their logical system to be useless as a means of making inferences
or providing demonstrations. Thus it is significant that in those passages where he
attributes the Philonian definition to the Stoics, Sextus then invokes this alleged
characteristic in an attack on some aspect of the Stoic theory of inference. For
example, consider the passages at PH 2.104 and at AM 8.244 where he assails the
Stoic doctrine of signs. 143 According to Sextus, the Stoics define the sign as an
142 See pages 407 and 479 for references. Long and Sedley take it that the ‘cohesion’ criterion is
the standard Stoic doctrine for a sound conditional; moreover, they think that “it probably had
the approval of Chrysippus” [Long and Sedley, 1990, 1.211).
143 Sextus claims not to reject altogether the existence of signs, but only those signs which the
‘dogmatic philosophers’ and ‘logical physicians’ call ‘indicative’ (evSeixtixov) (AM 8.156). On
the other hand, those signs called ‘commemorative’ (uxopvricmxov) are accepted by Sextus, since
The Megarians and the Stoics
485
antecedent axioma in a sound conditional, capable of revealing (exxaXuitxtxoc) the
consequent (AM 8.245; PH 2.104). Since there are three possible combinations of
truth values for a sound conditional, 144 the Stoics further stipulate that the sign
will be the antecedent of a sound conditional which begins with a true axioma
and ends with a true axioma (AM 8.248-50; PH 2.106). Clearly, nothing in this
definition of a sign commits the Stoics to the Philonian criterion for a sound con¬
ditional; however, Sextus claims that this is the criterion they accept (AM 8.247;
PH 2.105). In proceeding with his criticism, Sextus points out that, according
to the Stoics, the thing signified (aqpEtcoTov) is either ‘pre-evident’ (xpoSrjkov) or
‘non-evident’ (a8r|Xov) (AM 8.265; PH 2.116). But if pre-evident, then “it will not
admit of being signified, nor will it be signified by anything, but will be perceived
of itself’ (AM 8.267). On the other hand, if it is non-evident, then it cannot be
known that it is true, since if it were known, it would then be pre-evident (AM
8.267). Hence, although the truth-value of the antecedent is known to be true,
that of the consequent is necessarily uncertain. Therefore, the truth-value of the
conditional is uncertain, since the truth-value of the consequent must be known in
order to determine the truth value of the conditional (AM 8.268). Hence the Stoic
account of a sign is useless, since the soundness of a conditional with a non-evident
consequent is indeterminate (AM 8.268).
At AM 8.449, using a similar strategy, Sextus attacks the Stoic argument
schemata, and in particular, the first indemonstrable. 145 He intends to show that
an argument having this schema cannot in fact be a demonstrative argument,
and hence, is of no use in demonstrating a conclusion. According to the Stoics, a
demonstrative argument (dnoSeixTixoc Xoyoc) is an argument which is conclusive
(ouvaxTixo<;), has true premisses and a true conclusion, and deduces a non-evident
conclusion from pre-evident premisses (PH 2.140; AM 8.422). Starting with this
definition, Sextus proceeds roughly as follows (AM 8.449-52). Given any argument
for which the premisses are pre-evident, the conclusion is either pre-evident and
known, or it is non-evident and unknown. If the conclusion is pre-evident and
known, then according to the definition of a demonstrative argument, such an ar¬
gument is not demonstrative. On the other hand, if the conclusion if non-evident
and unknown, then the truth value of the conditional premiss is indeterminate.
For the antecedent of this conditional is the minor premiss of the argument, which
is pre-evident and known to be true, while the consequent is the conclusion of the
argument, which we are assuming to be non-evident and unknown. Hence, since
it cannot be determined whether the premisses are true, it cannot be determined
that the argument is demonstrative; therefore, in either case, the argument is not
he takes them to be among the ‘common preconceptions of mankind’ (xotu; xoivctic x£Sv avdpwTtuv
7 ipXii<j>emv) (8.157). Thus, Sextus is apparently attacking the conception as well as the Stoic
theory of indicative signs in these passages, but not the Stoic theory of commemorative signs.
144 Note that on each of the four accounts, not just on the Philonian definition, a conditional
with a true antecedent and false consequent will not be sound; moreover, on each account a
sound conditional may have either a true antecedent and true consequent, or a false antecedent
and false consequent, or a false antecedent and true consequent.
145 i.e., an argument with the schema ‘If p, then q; p; therefore q.
486
Robert R. O’Toole and Raymond E. Jennings
demonstrative. But if it is not demonstrative, then it is of no practical use as a
means of inference.
As Ian Mueller points out [Mueller, 1978, p. 23], if the truth-functional inter¬
pretation of the conditional is taken as the Stoic criterion, then no defence against
Sextus’ argument can be mustered on behalf of the first indemonstrable. On the
other hand, if the connexion interpretation (or the inclusion interpretation) were
taken as the Stoic definition, then Sextus’ argument would fail. It should be obvi¬
ous that the ascription of either the connexion criterion or the inclusion criterion to
the Stoics would, in addition to blocking Sextus’ criticism of the indemonstrables,
also nullify his objection, discussed above, to the Stoic doctrine of signs. With
respect to the criticism of the indemonstrables, Mueller expresses the point thus:
There is no way out of this situation, a fact that strongly suggests
that Sextus’ insistence on applying the truth-functional interpretation
to the conditional represents an argumentative device rather than an
accurate reflection of standard Stoic doctrine. If the first premise of an
undemonstrable argument expresses a stronger than truth-functional
connection between its component propositions, there is no reason why
the first premise can not be established independently of the conclusion
[Mueller, 1978, p. 23].
Mueller goes on to point out that the ascription of a strong interpretation to
the Stoic conditional “means that philosophically a great deal of weight must be
placed on the knowledge of necessary connections between propositions” [Mueller,
1978, p. 23]. It seems to us, however, that although Mueller’s point is correct,
his putting the matter in this way is somewhat misleading. Although we will not
argue for the point here, we would suggest that it is because their view places a
great deal of philosophical weight on a knowledge of necessary connections between
axiomata that one ought to ascribe a strong interpretation of the conditional to
the Stoics, and not the reverse.
A further point mentioned by Mueller is that many of the criticisms put forward
by Sextus in the course of his writings are directed against the possibility of there
being such knowledge of necessary connections as the Stoics suppose [Mueller,
1978, p. 23]. However, since it is not our intention here to defend Stoic doctrines
from the objections of Sextus Empiricus and other critics, these criticisms are
not a concern. What is of concern is to minimise the effects of Sextus’ claim
that the Stoics adopted the Philonian criterion for a sound conditional. One of
these effects, as has already been noted above, is Rist’s conjecture that it was
Zeno himself who opted for the Philonian view. Clearly this conjecture is at odds
with the interpretation we intend to put forward; we take it, however, that what
has been said about Sextus’ motives in ascribing the Philonian conditional to the
Stoics is sufficient to cast some doubt on his claim, and hence, to render Rist’s
conjecture doubtful as well.
There is yet another passage which would seem to indicate that the Stoics chose
the Philonian interpretation for the conditional. This passage occurs in Diogenes
The Megarians and the Stoics
487
Laertius, and hence, since Diogenes does not write in a polemical tone, one cannot
in this case invoke the sort of argument used against Sextus’ attribution. There
are, however, some doubts which can be raised against this ascription as well. The
passage in question is at 7.81. Here Diogenes says that according to the Stoics:
The true follows from the true, as, for example, ‘It is light’ from ‘It is
day’; and the false, from the false, for example, ‘It is dark’ from the
false ‘It is night’; and the true from the false, for example, ‘The earth
exists’ from ‘The earth flies’; but the false does not follow from the
true, for from ‘The earth exists’, ‘The earth flies’ does not follow.
The difficulty with taking this passage as an indication the Stoics adopted the
Philonian conditional (e.g. Mates [1953, 44nl4]) is that one is immediately con¬
fronted with an inconsistency in Diogenes’ account of Stoic logic. For at DL 7.73,
he reports that, according to Stoic theory, a true conditional is one in which the
contradictory of the consequent conflicts with the antecedent. This criterion is
precisely the definition of a sound conditional which Sextus described as the one
put forward by “those who introduce ‘connexion’ (PH 2.111). 146 So if one takes
the passage at DL 7.81 to indicate that the Stoics adopted the Philonian condi¬
tional, then it appears that the account from which Diogenes got his information
is inconsistent, since it attests to the adoption by the Stoics of two incompatible
definitions of a sound or true conditional. One possibility, of course, is that these
different views were predominant at different periods in the history of the Stoic
school. However, according to Hicks (DL 7.38nb), the source for the entire dox-
ography on the Stoics from DL 7.49 to DL 7.83 is Diodes of Magnesia, a scholar
of the first century B.C. considered by Mates “to have a fair knowledge of Stoic
logic” [Mates, 1953, p. 9]. Consequently, one would expect that the account given
in these passages would be fairly unified; moreover, one would also expect that
if criteria from different periods were included in this doxography, some mention
would have been made of the fact. Given these considerations, we would suggest
that rather than suppose an inconsistency in Diogenes’ source, one take the con-
nexivist definition as the standard Stoic criterion for the period covered by this
doxography and look for some other interpretation to explain the passage at DL
7.81.
Such an interpretation might be suggested by considering the examples intro¬
duced by Diogenes in this passage. It seems to us that the examples of a sound
conditional which he cites are all conditionals which would be sound according
to the connexivist criterion (and also, perhaps, according to the inclusion crite¬
rion). For instance, he illustrates the true-true case with the conditional ‘If it is
day, it is light’, and the false-false case with ‘If it is night, it is dark’. It seems
plausible that the Stoics might have taken the contradictory of the consequent in
these conditionals to be in conflict with the antecedent on the grounds that these
146 We are not unaware that Diogenes’ definition refers to a true (aXr]i3e<;) conditional, not a
sound (uyiec;) conditional. We have already argued, however, that in these contexts both ‘uytec’
and ‘dXrjdec’ should be understood as ‘proper’ or ‘genuine’ (See page 477).
488
Robert R. O’Toole and Raymond E. Jennings
states of affairs are related by a necessary causal sequence. Diogenes exemplifies
the false-true case with ‘If the earth flies, the earth exists’, and for this example
it is feasible that the Stoics might have invoked conceptual grounds to argue that
the contradictory of the consequent would conflict with the antecedent. What
we have in mind, then, is that the passage quoted above can be interpreted as
a demonstration of the point that it is possible for a sound conditional to have
either a true antecedent and true consequent, or a false antecedent and false con¬
sequent, or a false antecedent and true consequent. And in order to facilitate this
demonstration, the Stoics naturally presented examples of conditionals which they
took to be sound according to the connexivist criterion, since they adopted the
connexivist definition as their criterion for a sound conditional, as we are informed
at DL 7.73. For the case in which the antecedent is true and the consequent false,
which is to say, for the case where the conditional is not sound, Diogenes gives the
example ‘If the earth exists, the earth flies’. It would seem that the Stoics might
again appeal to conceptual considerations in order to say that in this example the
contradictory of the consequent does not conflict with the antecedent; moreover,
since this criterion determines an unsound conditional (DL 7.73), they could also
say on connexivist grounds that this conditional is not sound.
We have already suggested that, given the Stoic definition of a sign, Zeno’s
interest in the doctrine of signs would afford him reason to take a stance with
respect to the criterion for a sound conditional (see page 407). And we have also
put forward the view that the purpose of logic for Zeno’s wise man is to allow
him to make correct judgements about the connections between particular states
of affairs on the basis of his knowledge of the general causal principles governing
such connections (see page 422). Hence, we take it that Zeno’s interest in signs is a
manifestation of his general concern to draw out the implications of one’s actions
in accordance with the natural sequence of events. And since Zeno identified
the natural sequence of causation, which he called heimarmene with the logos or
rational principle of the universe (DL 7.149), and since he also identified the logos
itself with ‘necessity’ (Lactanius, Tertullianis SVF 1.160), our understanding is
that he saw these causal connections as necessary in the sense implied by these
identifications. Moreover, we propose that he chose the conditional construction
as the syntactical representation of these connections because such representation
is suggestive of the causal sequence of events. Hence, the Stoic use of the particle
‘el’ is technical and already implies a strong interpretation of the conditional since
it presupposes a necessary connection between the antecedent and consequent (see
page 423).
Therefore, it seems unlikely to us that Zeno would have adopted the view of
either Philo or Diodorus, for on either of these conceptions a conditional may be
sound even though there is no connection between its parts. Moreover, if the
invention of the connexion view can be attributed to Chrysippus, then it seems
evident that Zeno could not have opted for this definition since it would not have
been available to him. One might conjecture that he proposed an account of his
own, and if this were the case, then plausibly he introduced the inclusion criterion.
The Megarians and the Stoics
489
On the other hand, it is possible that he put forward a version which is completely
unrecorded, although it seems to us that this alternative is not so plausible as the
first one. In any event, although we shall pursue the matter no further in this
work, it would be of interest to explore the possibility that the inclusion definition
of a sound conditional was the criterion with which Zeno worked.
Consequence and conflict
The view of Mates and others notwithstanding, it would appear to be an open
question how one is to understand the use of the verb ‘pctx eTOa ’ ' n the passages
where it is used in the definition of a sound conditional by Sextus Empiricus (PH
2.111) and Diogenes Laertius (7.73; 77). The consensus among these commenta¬
tors is that it should be understood as ‘is incompatible’ where ‘incompatible “is
used in its ordinary sense,” which is to say, in the sense that two “incompatible
propositions cannot both be true, i.e., their conjunction is logically false” [Mates,
1953, p. 48]. On this view, then, a valid argument, according to the connex-
ivists, is such that it is not logically possible for both the contradictory of the
conclusion and the conjunction of the premisses to be simultaneously true. But
even leaving aside the difficulty of determining the Stoics’ understanding of ‘logical
possibility’, 147 there remain some etymological questions as to whether one ought
to accept this account as accurately reflecting the intension of ‘|i&x £TOtt ’ in Stoic
terminology.
To forestall possible objections to putting an etymological cast on the problem,
it would be useful to consider Mates’ criticism of Philip De Lacy for the latter’s
use of “weaving together” as a translation of the Greek term ‘oupTiXox/]’, which is
standardly translated as ‘conjunction’.
aupitXoxf), the technical term for conjunction, should not be translated
as “weaving together.” There is no virtue in employing etymological
translations for technical terms, since a term becomes technical pre¬
cisely by being dissociated from its etymological and other connota¬
tions and associated unambiguously with its denotation [Mates, 1953,
92n24],
Doubtless one can agree that it is never virtuous and perhaps always somewhat
fanciful to translate a technical term by summoning forth its etymological origins.
But it does not follow that in attempting to understand a term, the technical
meaning of which is either unclear or controversial, one ought to ignore its semantic
history.
In the present case, there are those who believe that ‘incompatibility’ designates
a non-truth-functional relation which exists between propositions. 148 They would
I47 Consider, for example, Gould’s comment that “it may be the case that [the] distinction
[between empirical impossibility and logical impossibility] had not, as a matter of fact, been
discerned in the Hellenistic age” [Gould, 1970, p. 81].
148 See, for example, E..J. Nelson [1930, esp. pp. 440-43]; and R.M. Stopper, [1983, pp. 281-86].
490
Robert R. O’Toole and Raymond E. Jennings
criticise Mates’ and those who agree with him on the grounds that, according
to his characterisation, it would turn out that an impossible proposition would
be incompatible with any proposition, even itself. This result is not in accord
with their logical intuitions. The relation which Mates describes is, on their view,
more aptly designated as incompossibility . li9 What is relevant to this controversy,
however, is the question of how the Stoics understood the meaning of ‘[idtx £Tal ’>
which, as has been noted, is standardly translated as ‘incompatible’. It is here
that one can look to etymology for assistance.
The primary meaning for pax 20 '®® 1 ) the infinitive form of the verb, is to fight or
to battle or to war. Now, one would hardly want to translate the term ‘[idxETou’ in a
logical context as ‘fights’ or ‘battles’ or ‘wars’. To the modern logician, not only do
they seem somewhat fatuous as a description of a relation between propositions,
but also they seem rather out of place in a logic treatise. Nevertheless, these
renderings would seem to reflect more faithfully the etymology of ‘pdxexoti’ than
does the translation ‘is incompatible’, at least where the latter is understood as
Mates understands it. Probably ‘conflicts’ is just the right compromise. It is
bloodless enough for a logic book, yet it remains faithful to the etymological origins
of the Greek term, more so, it would seem, than ‘is incompatible’.
Now with this translation in mind, consider the thesis that the Stoics understood
this notion of conflict in terms of ‘incompossibility’, where this term is taken in the
sense that two propositions are incompossible just in case it is not possible that
they both be true. It has been pointed out that, according to this characterisation,
it is a sufficient condition for two propositions to be incompossible if one of them
is necessarily false. Hence, the propositions ‘All triangles have four sides’ and
‘Chrysippus is the greatest of Stoic logicians’ would be incompossible. Would the
Stoics have considered these propositions to be in any sense ‘in conflict’? It is
difficult to see how anyone would suppose them so. On the other hand, consider
the propositions ‘All triangles have four sides’ and ‘All triangles have five sides’.
These propositions are clearly incompossible because both are impossible, but it is
also clear that they are related in such a way that if one is affirmed, then the other
must be denied. This relation, moreover, is independent of the truth values or the
modal status of the individual propositions. It is this sort of relation which the
critics of Mates’ view appear to have in mind as the proper meaning of the term
‘incompatibility’. And, one might assume, it is also what they would expect that
the Stoics had in mind when the latter spoke of propositions or states of affairs
being ‘in conflict’.
149 M.R. Stopper quoting from a paper by Mauro Nasti de Vincenti (“Logica scettica e impli-
cazione stoica,” in Lo scetticismo antico, ed. G. Giannantoni, Naples, 1981.), writes that “‘P’
conflicts with ‘Q’ just in case ‘P’ and ‘Q’ are not compossible,” and he symbolises this definition
thus [Stopper, 1983, p. 284]:
(A3) C(P,Q) <-> L->(P A Q)
He goes on to say that (A3) has “some strange consequences.” For example, “any impossible
proposition is incompossible with any other proposition whatsoever” [Stopper, 1983, p. 285].
Trje Meyapiatvc av8 xr]£ Stoicc
491
8.2 The conditional and inference
Validity and conditionalisation
The so-called principle of conditionalisation is presented in several places by Sextus
Empiricus as a Stoic criterion for a valid argument. 150 As it is framed by the Stoics,
this canon states that an argument is conclusive 151 corjeveep lie; ^oppeonovdivy
<^ov8momX lc ctouvS (Gyiec : PH 2.137) or true (aXqOEc;: AM 8.417): that is,
the conditional which has the conjunction of the premisses as antecedent and the
conclusion of the argument as consequent. As an example, Sextus presents the
following case at PH 2.137. The argument
(1) If it is day, it is light;
it is day;
therefore it is light
has as its corresponding conditional the following:
(2) If (it is day, and if it is day, it is light), it is light.
The application of the principle here is the assertion that since the corresponding
conditional is sound, the argument is valid. According to Sextus’ account at AM
8.111-12, it was agreed among the ‘dialecticians’ that a conditional axioma is sound
whenever its consequent axioma ‘follows’ its antecedent axioma. Hence, to say that
(2) is sound is just to say that its consequent, which corresponds to the conclusion
of (1), indeed follows its antecedent, which corresponds to the conjunction of the
premisses of (1). It seems evident that if the Stoics wished to attribute such a
property to a valid argument, then they must have assumed that the relation of
‘following’ (axoXouflta), which they took to be the relation holding between the
antecedent and consequent in a sound conditional, was the same relation holding
between the premisses and conclusion of a valid argument (cf. PH 2.113).
Taking into account the debate over the sound conditional discussed in Section
8.1, as well as the principle of conditionalisation, one would expect that there
would have been recorded as many distinct conceptions of a valid argument as
there were accounts of a sound conditional. This does not, however, seem to
be the case. Other than the conditionalisation principle itself, there appears to
be no mention in the fragments of a criterion for a valid argument except the
one implied by Diogenes Laertius at 7.77. In this passage Diogenes presents the
following characterisation of an invalid argument.
150 e.g., AM 8.415; PH 2.113, 137. See Mates ([1953], 74-77) for a discussion of this principle. As
Mates points out, this principle need not be taken as defining the Stoic conditional, but merely
as a characterising a property common to all valid arguments.
151 In some places (e.g., PH 2.137, 146) Sextus uses ‘ctuvoixtixoc’ and ‘dovvccxxdz’ for ‘conclusive’
and ‘inconclusive’ (or ‘valid’ and ‘invalid’), whereas at other places (e.g., AM 8.429) he uses
‘jtEpavxixoc’ and ‘aitepavxoc’. Hence, as Mates indicates in his glossary [Mates, 1953, pp. 132-
36], these terms appear to be interchangeable.
492
Robert R. O’Toole and Raymond E. Jennings
And of arguments some are conclusive (valid) and some inconclusive
(invalid). Inconclusive are those in which the contradictory of the
conclusion does not conflict with the conjunction of the premisses. 152
Although it is not explicitly stated, this characterisation would seem to imply that
a valid argument is one in which the contradictory of the conclusion is in conflict
with the conjunction of the premisses.
In addition to the above account which implies a criterion for a valid argument,
Diogenes also reports the following Stoic criterion for a sound conditional.
So, then, the true conditional axioma is one in which the contradictory
of the consequent conflicts with the antecedent, as in this example: ‘If
it is day, it is light’ (DL 7.73). 153
It is evident that the criterion for a sound conditional described in the passage
at DL 7.77 is identical to the one which Sextus Empiricus reports at PH 2.111.
This is the criterion proposed by “those who introduce ‘connexion’ or ‘coherence’
(auvdpxr|aic)” as a condition on the relation of following between the antecedent
and consequent of a sound conditional. It was mentioned earlier that this standard
has been ascribed by several modern commentators to Chrysippus himself (see
page 479). In light of his influence on the development of Stoic logic, it is probable
that if this ‘connexivist’ view was indeed the one he advocated, then it would have
been the one accepted by the Stoa.
The formulation of the connexivist criterion leaves no doubt that its adoption
would commit the Stoics to a strong interpretation of the criterion for a sound con¬
ditional axioma. Thus it seems plausible that for the Stoics the term ‘axoXouflEiv’
expressed a real connection or coherence between the antecedent and consequent,
and, in some sense, a necessary relation between them. Since the conditionali-
sation principle implies that the same relationship holds between the premisses
and conclusion of a valid argument, we can infer that such a connexion obtained
between them as well.
Now in accordance with the conditionalisation principle, the ubiquitous Stoic
example
(3) If it is day, it is light;
it is day;
therefore it is light
would be valid just in case the following conditional were sound:
152 tSv 8e Xoyov oi pev eicnv ootepavToi, oi 8e itEpavtixoi'. aitepavTOi pev Cv to avTixeipevov Tfjc
eitupopac ou paxEtai Tfj 8ia t£5v XTjppaxrov aopr.Xoxfj,
153 auvr)ppevov ouv aXrpElEC ecttiv ou to avTixdpevov too XrjyovToc pax E1:al rjyoupivu, oTov ‘ei
ripepa ectti, (pfik ectti.’
The question concerning the interpretation of the Greek term ‘paxsvxi’, which we have rendered
as ‘conflicts’, has already been discussed (see page 489). We have argued that the notion of
conflict which the Stoics had in mind requires some degree of common content between the
axidmata in this relationship.
The Megarians and the Stoics
493
(4) If (it is day, and if it is day, it is light), it is light.
And in accordance with the description of a sound conditional given by Dio¬
genes Laertius at 7.73, (4) would be sound just in case the contradictory of its
consequent were in conflict with its antecedent. Thus, in conformity with the con-
ditionalisation principle and the description of a sound conditional presented by
Diogenes, (3) would be valid just in case the contradictory of its conclusion were
in conflict with the conjunction of its premisses. This would seem to suggest that
the characterisation of a valid argument given by Diogenes at 7.77 is derived from
an application of the connexivist notion of a sound conditional to the principle of
conditionalisation.
There are, however, difficulties with this proposal. The first objection is that
there are the passages in Sextus Empiricus (PH 2.104; AM 8.245) which seem to
indicate that the Stoics adopted the Philonian account of a sound conditional (cf.
[Mates, 1953, p. 43]). A further objection is that both Mates [1953, p. 60, 75] and
Bochenski [1963, p. 97] cite passages at PH 2.137 and AM 8.415 to support the
thesis that the conditionalisation principle required a ‘Diodorean-true’ conditional.
The views of the Dialecticians Philo and Diodorus have been discussed earlier (see
page 479), however, a brief summary of their views might be in order for the
present. According to Philo, then, a conditional is sound whenever it does not
have a true antecedent and false consequent (PH 2.110; AM 8.113). According to
Diodorus, on the other hand, a conditional is sound if it neither was capable nor is
capable of having a true antecedent and false consequent (PH 2.110; AM 8.115).
In replying to the first objection one probably cannot deny that the texts appear
to support the view that the Philonian account gained some measure of acceptance
among the Stoics. One might point out that acceptance of this account was by no
means unanimous, as the passage at AM 8.245 indicates. And even if this was the
view chosen by many Stoics, the debate continued. 154 If it were the case that they
did opt for the Philonian criterion, then one would expect that applications of the
conditionalisation principle would reflect that fact. But we believe that a more
telling reply would be to point out the inconsistencies in Sextus’ various reports.
First, the adoption of the Philonian truth conditions would seem to be in conflict
with the reported wide acceptance of the doctrine that an argument is valid when
and only when its corresponding conditional is sound. Since there is no neces¬
sity in the relation between the antecedent and consequent of a sound Philonian
conditional, it is hard to see how such a conditional could underwrite the validity
of its corresponding argument. Against this reply, one might propose, as Josiah
B. Gould does [Gould, 1974, p. 160], that the advocates of the Philonian view
perhaps invoked the Diodorian truth conditions in applications of the conditional¬
isation principle. One might point to those passages cited by Mates and Bochenski
wherein it appears that the Stoics had the Diodorean conditional in mind when
154 xp(oEic; §e tou uyiouc ctuvtjupievov jioXXac pcv xai aXXac eTvai tpotcriv, piiav 8’ e? cmaaGv Gxapxeiv,
xai tauTT)v oux opioXoyov, xr)v anoSo-drioopLevriv (AM 8. 245).
494
Robert R. O’Toole and Raymond E. Jennings
they framed this principle. This approach, however, has its own problems. This
interpretation of the texts mentioned above would seem to be inconsistent with
the account of the criteria for invalidity referred to by Sextus at PH 2.146-51 and
AM 8.429-34.
In this account Sextus reports that the Stoics deemed an argument invalid ac¬
cording to a list of four criteria. These are: having premisses and conclusion which
are incoherent with one another, having redundant premisses, being propounded
in an invalid form, and having a deficient premiss. A problem arises when one at¬
tempts to square the first criterion on this list with the proposal that the principle
of conditionalisation required a Diodorean-sound conditional. As a consequence
of this proposal, an argument such as the following would appear to be valid:
(5) If Dion is walking, he is moving;
but wheat is being sold in the market;
therefore, the elements of existents are without parts.
One would be committed to judge (5) as valid if the following, which is its
corresponding conditional, were Diodorean-sound:
(6) If (wheat is being sold in the market, and if Dion is walking, then
he is moving), then the elements of existents are without parts.
But (5) could not be considerd valid according to the criterion which prohibits
incoherent (SidtpTrjou;) arguments from being valid. The problem, therefore, is that
if the principle of conditionalisation requires a Diodorean-sound conditional and
if (6) were Diodorean-sound, then (5) would be valid, contrary to the criterion
for invalidity mentioned above. On the other hand, if this criterion for invalidity
were to prevail, then (5) could not be valid and the conditionalisation principle
could not require a Diodorean-sound conditional, provided that (6) is Diodorean-
sound. Thus, if (6) is Diodorean-sound and (5) is not valid, then it is not clear
that one ought to accept the thesis that the principle of conditionalisation requires
a Diodorean-sound conditional.
Now it is apparent that Diodorus would have been committed to the soundness
of (6) merely because of the modal status of its consequent, for according to Sextus
Empiricus (PH 2.111), Diodorus would deem the following conditional to be sound.
(7) If it is not the case that the elements of existents are without parts,
then the elements of existents are without parts.
It was determined earlier (see page 481) that the consequent of this conditional
would have been considered necessarily true according to the Diodorean view of
necessity. Hence, the corresponding conditional of (5) would be Diodorean-sound
merely because, according to Diodorus, its consequent could have been neither false
nor false. That is, (6) neither was nor is capable of having a true antecedent and a
false consequent, since it neither was nor is capable of having a false consequent. It
The Megarians and the Stoics
495
would seem apparent, then, that (6) is Diodorean-sound. It is not clear, therefore,
that one need accept the contention that the principle of conditionalisation requires
a Diodorean-sound conditional.
In view of the foregoing arguments, neither of the objections considered is de¬
cisive against the proposal that the connexivist standard was the criterion for a
sound conditional which prevailed in the early Stoa. The formulation of this ac¬
count, which was put forward by “those who introduce ‘connexion’ or ‘coherence’”
as a condition on the relation of following between the components of a sound
conditional, would seem clearly to have committed the Stoics to a strong interpre¬
tation of the relationship between the antecedent and consequent of a conditional
axidma. Thus the Stoics would seem to have understood the term ‘to follow’
(axoAouffcIv) as expressing a necessary relation, in the appropriate sense of neces¬
sary, not only between the antecedent and consequent of a sound conditional, but
also, as a result of the connection between a valid argument and its corresponding
conditional, between the premisses and conclusion of a valid argument.
General conditionals
According to Josiah Gould, as we have seen, Chrysippus thought that one could
generalise on the observed relations between different types of states of affairs or
events and express these generalisations as conditional statements [Gould, 1970,
pp. 200-201]: see page 426). What is required, then, is an account of such
general conditionals, and clearly the relationship between singular and general
conditionals will need to be sorted out. Unfortunately, there are very few examples
of such general conditionals in the extant texts; however, the few that there are
would seem to be sufficient to indicate the pattern. An example occurring in
Cicero is as follows: “If anyone ( quis ) was born at the rising of the dogstar, he
will not die at sea” (De fato 12). Another example occurs in Sextus Empiricus
where he informs us that, according to the writers on logic, “the definition ‘Man
is a rational, mortal animal’, although differing in its construction, is the same
in meaning as the universal (xordouAixov) ‘If something (xl) is a man, that thing
(exelvo) is a rational, mortal animal’” (AM 11.8). Other examples are available,
but the pattern for the general conditional seems apparent. Evidently, the subject
of the antecedent clause is expressed by an indefinite pronoun, and though it is
not clear in the Latin example, the Greek example would seem to indicate that
the subject of the consequent clause having anaphoric reference to the subject of
the antecedent clause is also an indefinite pronoun. 155
Now consider the example of a singular conditional from AM 8.305 and what
we might call its ‘associated’ general conditional, the latter being constructed on
the pattern determined above. The singular conditional is If Dion is walking, Dion
is moving, and the associated general conditional would be If someone is walking,
155 It should be noted here that although ‘exeTvo’ would normally be classed as a demonstrative
pronoun, it seems evident that in constructions such as this where it serves as a relative pronoun
with anaphoric reference to an indefinite pronoun, it must be taken as an indefinite relative
pronoun.
496
Robert R. O’Toole and Raymond E. Jennings
he (or that one) is moving. In his paper “Stoic Use of Logic,” William H. Hay has
suggested that what we have here is, in effect, a universally quantified conditional
and an instantiation of it [Hay, 1969, 151n22]. If this assessment is correct, then it
would evidently imply not only that the Stoics used general conditionals in place
of statements using ‘all’, 156 as well as employing a rule of instantiation for deriving
singular conditionals from general ones, but also that their logic cannot be viewed
on this account as simply a logic of propositions. The suggestion expressed by
Hay raises a difficulty which is communicated by Charles Kahn in the following
dilemma:
Either Stoic logic is based solely on the propositional connectives, and
then it is epistemically sterile ... Or else it involves generalized condi¬
tionals and a rule of instantiation, but then it is defective as logic since
we are left without any account of the quantified conditional [Kahn,
1969, p. 164].
Now we believe that what Khan has in mind here in setting out the first horn of this
dilemma is a propositional logic with a classical truth-functional interpretation of
the propositional connectives. It is worth noting that the classical interpretation
of the connectives is only one of many possible interpretations which might be
assigned to them; hence, given an appropriate interpretation, a propositional logic
need not be so barren as Kahn envisages. In any case, it seems evident that Stoic
logic was not a classical propositional logic, and could not, therefore, be viewed as
‘epistemically sterile’ on the assumption that it was; moreover, it also seems clear
that the Stoics themselves did not consider their logic to be so. Thus, we would
reject the first horn of the dilemma. As for the other horn, we find it difficult to
agree that the Stoic system was ‘defective’ as logic because it lacks an account
of the quantified conditional. Kahn writes that “it is time to return to a more
adequate view of Stoic logic within the context of their theory of language, their
epistemology, their ethical psychology, and the general theory of nature” [Kahn,
1969, p. 159]. This suggestion would seem to imply that Stoic logic be assessed on
it own terms and not as an attempt at constructing a modern formal system. In
putting forth his criticism, Kahn seems to be ignoring his own reproach. At any
rate, it may be that one can give an account of general conditionals which justifies
the inference from general conditionals to singular or particular conditionals, and
do so without invoking universal quantifiers and a rule of instantiation.
A general conditional, as has been noted above, seems to be signified by a
conditional sentence having an indefinite pronoun in the subject position of the
antecedent and an indefinite pronoun having anaphoric reference to the subject of
the antecedent. It seems evident that the general conditional would be true just
in case every associated particular conditional which has either a demonstrative
pronoun or a name in the subject position, is true. Thus the general conditional
156 Mates has pointed out that “nowhere in the rather elaborate classification [of propositions]
is any provision made for universal affirmative propositions, that is, for propositions beginning
with ‘all’” [Mates, 1953, p. 32],
The Megarians and the Stoics
497
may have been viewed as the conjunction of its associated particular conditionals.
If the Stoics were to have allowed the inference of the conjuncts of a conjunction
without an explicit rule of conjunction elimination, then this might explain why
they seem to have supposed that one could infer the particular conditional from
the general conditional without a rule of universal elimination.
9 FORM IN STOIC LOGIC
Over the long history of what is referred to as Stoicism, there was no doubt much
unrecorded even unnoticed variability in metalogical doctrine. No doubt many
distinctions which we now take for granted were ‘beneath the level of specificity
of their intentions’. Nor is there any reason to suppose that the Stoics surpassed
twentieth-century philosophers in their awareness of the degree of indeterminacy of
their adopted theoretical language, or of their prospects for success. And no doubt,
their approach consisted, to some extent, in talking in order to find out what they
were talking about. So there might be little point in looking for a mathematically
precise account of their doctrines, even if the historical records were much more
complete than they are. In fact their intellectual environment was so different from
our own as to have long since rendered their semantic space largely inaccessible
to us. We simply cannot reconstruct, let alone reproduce, the effects that their
theoretical vocabulary could have been counted upon to have. The best we can
hope for is an illusion of precise positive understanding. We can, however, take
some precautions against particular misunderstandings of their project. More
specifically, and for all likely purposes, most usefully, we can take account of ways
in which their logical culture and methods differed from our own, and take due
note of the superficiality of apparent similarities between their approach and ours.
Positively, we can give more reliable shape to Stoic logical theory by using our
own richer notational resources to approximate their conceptions and engage their
subject matter. And we can try to triangulate their position by considering what
theoretical resources lay nearly within their reach.
In this section we illustrate the difficulties by a detailed consideration of the
Stoic notion of (SieCsoypEvov) (disjunction) in relation to the question as to whether
Stoic logic can be regarded as formal in the twentieth-century use of the word.
9.1 First blush
The superficial similarities of the indemonstrables to a set of natural deductive
rules may tempt the unwary to a reconstruction in the language of twentieth-
century formal systems, to define the elements of the language, the atoms, the
connectives, the well-formed formulae, and then to introduce the rules for ex¬
tending proofs. This would be to suppose that the Stoics viewed the connective
vocabulary of the indemonstrables as having uniform logical status. A closer ex¬
amination would reveal that the supposition was unwarranted. They seem to have
been interested in vocabulary whose correspondents had, for them, some degree
498
Robert R. O’Toole and Raymond E. Jennings
of physical eclat. So they were more interested in disjunctions than conjunctions,
and more interested in conditionals, than in negations. In fact, although there
seems to have been some unclarity on this score, their focus was primariy upon
relationships, conflict and consequence, for example, and only secondarily upon
the vocabulary that was used to distinguish them.
A related temptation would be to suppose that because a connective would ad¬
mit an introduction or an elimination rule that coincides with an indemonstrable,
that must have been the connective that the Stoics had in mind. Consider first
the accepted doctrine that indemonstrables [IV] and [V] rely upon the exclusive
disjunction of
a Y p
1 0 1
1 1 0
0 1 1
0 0 0
I. M. Bochenski:
... out of the fourth and fifth indemonstrables which were fundamental
in Stoic logic, we see that exclusive disjunction (matrix ‘0110’) was
meant. ([Bochenski, 1963], 91)
W. and M. Kneale (on Galen’s remark that ‘Either it is day or it is night’ is
equivalent to ‘If it is not day it is night’):
Possibly his expression is loose and he means to say that the disjunctive
statement is equivalent to the biconditional ‘It is not day, if and only
if, it is night’. For the assertion of such an equivalence would indeed be
in keeping with the Stoic doctrine of disjunction, provided always that
the conditional is understood to convey necessary connection. ([Kneale
and Kneale, 1962b], 162)
Benson Mates:
Two basic types of disjunction were recognized by the Stoics: exclusive
and inclusive. Exclusive disjunction (SisCeuypevov) was most used, and
is the only type of disjunction which occurs in the five fundamental
inference-schemas of Stoic propositional logic. ([Mates, 1953], 51)
Lukasiewicz:
It is evident from the fourth syllogism that disjunction is conceived of
as an exclusive ‘either-or’ connective. ([Lukasiewicz, 1967], 74)
The Megarians and the Stoics
499
Ian Mueller:
‘The first or the second’ is true if and only if exactly one of the first
and the second is true. (In modem logic it is customary to use ‘or’
inclusively, and hence to substitute ‘at least’ for ‘exactly’ in the truth
conditions for disjunction. The fourth indemonstrable argument shows
that disjunction is exclusive in the Stoic system.) ([Mueller, 1978], 16)
9.2 Some evidence
All of these authors cite ancient sources for this account, among them, Cicero,
Gellius, Galen, Sextus Empiricus and Diogenes Laertius. Their accounts are the
following:
Cicero:
There are several other methods used by the logicians, which consist
of propositions disjunctively connected: Either this or that is true; but
this is true, therefore that is not. Similarly either this or that is true;
but this is not, therefore that is true. These conclusions are valid
because in a disjunctive statement not more than one [disjunct] can be
true. 157
Gellius:
There is another form which the Greeks call SieCeuyuevov and
we call disiunctum. For example: ‘Pleasure is good or evil or it is
neither good nor evil.’ Now all statements which are contrasted ought
to be opposed to each other, and their opposites, which the Greeks
call avTtxeipeva, ought also to be opposed. Of all statements which are
contrasted, one ought to be true and the rest false. 158
Galen:
... the disjunctives have one member only true, whether they be com¬
posed of two simple propositions or of more than two. 159
157 Topica, 14.56-7. Reliqui dialecticorum modi plures sunt, qui ex disiunctionibus constant:
aut hoc aut illud; hoc autem; non igitur illud. Itemque: aut hoc aut illud; non autem hoc; illud
igitur. Quae conclusiones idcirco ratae sunt quod in disiunctione plusuno verum esse non potest.
158 Nodes Atticae , 16.8. Est item aliud, quod Graeci 8ie£euym-£ vov nos ‘disiunctum’
dicim.us. Id huiuscernodi est: ‘aut malum est voluptas aut bonum neque malum est\ Om¬
nia autem, quae disiunguntur, pugnantia esse inter sese oportet, eorumque opposita, quae
otviixd(Jieva Graeci dicunt, ea quoque ipsa inter se adversa esse. Ex omnibus, quae disiunguntur,
unum esse verum debet, falsa cetera.
159 Inst. Log., 5.1. ... xc5v sv (jlovov e^ovtcov aXrydEC, av t' ex 5uoTv a^tco^aTCov
aitXaSv 5 av t' ex xXeiovwv auYxsr)Tai. (The translation is that of Kieffer [1964].)
500
Robert R. O’Toole and Raymond E. Jennings
Sextus Empiricus:
... for the true disjunctive announces that one of its clauses is true,
but the other or others false or false and contradictory. 160
Diogenes Laertius:
A disjunction is [a proposition] conjoined by means of the disjunctive
conjunction ‘either’ (qxoi) . For example, ‘Either it is day or it is night.’
This conjunction declares that one or the other of the propositions is
false. 161
9.3 The question of arity
The first point to attend to is that three of the five authors admit disjunctions of
more than two disjuncts, while two illustrate the construction with two-member
disjunctions. No great importance is attached to this by the commentators, and it
is unclear whether none of them thinks it significant. There need, of course, be no
great importance in the fact that the earliest and the latest of the sources quoted
above define disjunction specifically with reference to two-termed disjunctions. In
Diogenes’ example, it may only be because the illustration is two-termed that the
last comment is framed as it is. It is reasonable to surmise that neither Cicero nor
Diogenes Laertius would have precluded three-term or four-term disjunctions, and
that their account would coincide with those of Gellius, Galen and Sextus Em¬
piricus, according to which, in the three-term case, the disjunction is true if and
only if exactly one of its disjuncts is true. Since none of the modern commentators
explicitly addresses the issue of arity, one might have assumed that that is their
view of the matter as well. Bochenski [1970, p. 91] mentions the greater general¬
ity of Stoic conjunction ‘the [conjunctive] functor was defined by the truth-table
‘1110’ [sic] as our logical product (only an indeterminate number of arguments
was meant)’, and one may assume that his omission of the corresponding remark
about rjxoi is an oversight. But some explain three-member disjunctions as though
they nested a two-member disjunction. Commenting on the form:
Either the first or the second or the third; but not the first; and not the second; therefore the
third
which Sextus attributes to Chrysippus, the Kneales [1962a, p. 167] surmise:
Here, it seems, we must think of the words ‘the second or the third’
as bracketed together in the disjunctive premiss; for the conclusion
160 PH 2.191. to Y&P byiec Sie^euyuevov ErcaYYsXXETai ev tQv ev ctuxB uy'EC eTvou, to 8e Xoraov
fj Ta Aoutd <]>e08ot; yj ^eu8fj ^exa pLocx^C-
161 DL 7.72. 5ie£euYnivov 8e eaxiv 6 utco xou «yjxot» Sia^euxxixou auv8eap.oi> 8te^euxxat, otov
«Y)TOi rurepa eaxT fj v6£ eaxiv.» ercaYYsAAeTai § 6 auv8eap.o<; ouxoc to exepov x65v d^ta>p.dxa>v
c)>e08oc eTvai.
The Megarians and the Stoics
501
can then be obtained by two applications of indemonstrable 5. If this
procedure is correct, the disjunction may be as long as we please, since
the conclusion can always be proved by a number of applications of
the same indemonstrable.
But though bracketing will have the required effect in the case of the fifth in¬
demonstrable, its effect will be quite other in the case of the fourth; for correctly
inferring from the truth of the first disjunct the falsity of the disjunction of the
second and third will not then let us infer the falsity of the third from the truth
of the second: the disjunction of the second and the third may be false because
both the second and the third disjuncts are true. The conclusion must be that
although we can in isolated instances treat three-term disjunctions as nestings,
nevertheless if we are to give a unified account of Stoic disjunction, we may never
understand three-term disjunctions as understanding the second and third to be
implicitly bracketed. Brackets are simply not permitted. If this arity -free account
is the correct and most general account of the Stoic notion of disjunction, several
observations may be made: first that were we to symbolise such a connective it
would be unambiguous and natural to do so in prefix notation as:
V<r(ai,... ,Q„)
where the subscript a serves to make the Stoic connection explicit. For the (rjxot)
of Greek, like the ‘or’ of English, is not specifically a binary connective, and the
Stoic practice of representing sentences by nominals (to Tipoiepov, to Seuxepov, to
T pLTOv, the first, the second, the third) tends to mask the distinction which, when
in a philosophical set of mind, we implicitly make in English between, say, a list of
three nominals composed with ‘or’ and a three-term disjunctive sentence. In the
former case, we do not, indeed cannot think of the or-list of two of the names as
forming a new name and that disjoined to the third. In ordinary English we are
not required to think of the or-composition of three sentences in this way either.
No rules of well-formedness force us to parse a three-clause sentence composed
with ‘or’ into a two-clause sentence one of whose clauses is a disjunction. Except
for the exclusivity, the Stoic construction
qToi to xpoTEpov fj to SeuTepov f) to TptTOV
(Either the first or the second or the third)
alternatively,
7}TOL to a r) to p fj to y
is more like the syntax of ordinary Greek than the modern symbolization
£*i V (a 2 V q 3 )
502
Robert R. O’Toole and Raymond E. Jennings
is like the syntax of ordinary English, since to repeat the ‘either’ to express the
inner parenthesis would be stilted and unidiomatic. Now, to be sure, we could
abbreviate a modern n-term exclusive disjunction analogously by:
a„)
since exclusive disjunction is an associative operation. But although the ambiguity
is not vicious, we would normally understand such a formula as associated to the
left or to the right, since Y is a binary connective, and well-formedness requires it.
That modern exclusive disjunction is a binary truth-function and that the Stoic
notion had no fixed arity should not be lost sight of when comparing the two. It
will serve to remind us that the truth conditions of the two constructions are not
in general the same, a fact upon which none of the modern commentators seems
to have remarked. Consider as example the exclusive disjunction:
Y((2 + 2 = 4), (2 + 3 = 5), (2 + 4 = 6)).
When it is disambiguated into, say:
(2 + 2 = 4) Y ((2 + 3 = 5) Y (2 + 4 = 6)),
it becomes evident that since the second disjunction is false (since both of its
disjuncts are true) and the first disjunct is true, the whole disjunction is true in
spite of (or rather because of) the fact that all its disjuncts are true. The Stoic
disjunction:
V„((2 + 2 = 4), (2 + 3 = 5), (2 + 4 = 6))
is false, since more than one of its disjuncts are true. Since Stoic disjunction has
no fixed arity, it would be suitable to regard it as a kind of restricted propositional
quantifier, having, in prefix notation, the reading
Exactly one of the following is true:
Since exclusive disjunction is commutative and associative, a quantifier reading
would be suitable for it as well. But as a simple induction will demonstrate, its
quantificational rendering would be:
An odd number of the following are true:
The two kinds of disjunction will, of course, coincide on the two-clause case, but
will coincide for no n-clause case for n < 2. A three-clause exclusive disjunction,
for example, will be true if and only if either exactly one or exactly three clauses
are true, as will a four-clause exclusive disjunction. A five- or six-clause exclusive
disjunction will be true if and only if either exactly one or exactly three or exactly
five disjuncts are true, and so on.
A valid Stoic disjunction of two terms would disjoin a sentence a with a sentence
equivalent to the negation of a. A true n-term Stoic disjunction would disjoin n
The Megarians and the Stoics
503
finite state descriptions. As an example, imagine the formulation of a row of a
truth table, that is, the effect of conjoining propositional variables or negations of
propositional variables accordingly as l’s or 0’s appear under them in that row. A
valid Stoic disjunction in m independent variables would be equivalent to the 2 m -
term disjunction of the formulations of the rows of a table displaying all possible
combinations of their truth values. Particularly if, as some of the early sources
suggest, the Stoic notion of disjunction was that of an intensional operation, a
sentence of the form
fjxoi TO TlpOTEpOV fj TO ScUTEpOV f) TO TplTOV f) TO TETCipTOV
(Either the first or the second or the third or the fourth)
given such a technical use of fj' would assert that the four sentences bore to one
another a relationship akin to the relationship of the formulations of rows of a
two-variable truth table:
fiTOi {p A q) fj (p A -.g) fj (->p A q) fj (^p A -><?)■
As a consequence of this, if we seriously adopt the view that the disjunction that
Chrysippus had in mind in the fourth indemonstrable is the present day 0110
disjunction, then the Stoics really had at least two different kinds of disjunction
represented by the same piece of notation in their logical system. And having
come this far, we could admit no grounds for regarding the disjunction of the fifth
indemonstrable as anything but a third sort, namely 1110 disjunction. The more
plausible account would be that they had one sort of disjunction in mind, namely,
the disjunction of no fixed arity which happens to resemble 0110 disjunction in
the two-term case.
9-4 The consequences for their idea of form
The standard notion of form as applied to propositional argument schemata fol¬
lows these lines: let F be the set of sentences of a language L and S an argument
schema expressed with constants of the language L and metalogical variables of
the metalanguage ranging over F. Then the argument form Fs associated with S
is the set of arguments which can be generated from S by uniform substitution of
sentences of F for metalogical variables in S. This notion of form depends upon
a fixed meaning for the constants of L. In the propositional case, for example, we
assume that -i, A, V, — > and so on do not change their meanings as we uniformly
substitute sentences for the metalogical variables flanking them. We do not ac¬
count ‘a ; therefore a or p ’ an invalid form because from the sentence ‘You may
go or you may stay’ it follows that you may stay. We say rather that ‘You may go;
therefore you may go or you may stay’ is not of the form ‘a; therefore a or P’. We
might retreat, if pressed, to the claim that they share grammatical but not logical
form. Or we might admit the argument to the form but insist that the conclusion
must then be understood as a disjunction, from which ‘You may stay’ does not
504
Robert R. O’Toole and Raymond E. Jennings
follow. It seems certain that the Stoics never articulated a notion of argument
form in these or equivalent terms. But, if we are to take into account the totality
of evidence from early sources, according to which SleCeuypevov was understood
in something like the quantificational reading outlined earlier, and the generally
held view that they gave to the particle rjioi a technical meaning accordingly, then
the Stoic schema [IV]:
fjroi to Ttpotepov f| to SeuTepov
TO §E TipOTepOV
oux apa to SeuTEpov).
Either the first or the second;
But the first;
Therefore not the second.
is insufficiently general to capture the inferential force of the connective rj in their
technical sense. The three-term ^-disjunction is not obtainable from the two-term
^-disjunction by substitution of a two-term a-disjunction for one of the original
disjuncts. Some such schema as:
V<r(<3:1 j • • • j £*»>••■> Ot n )
<*j (1 < 3 < n);
.'. ~'(%k (1 < k < n){k ± j)
would be required. So if the Stoic notion of disjunction was as general as the
early commentators suggest, and we are to judge their conjectured position by
standards of any rigour, then we must conclude that their understanding of the
role of the fourth indemonstrable schema was something other than that of speci¬
fying a form in the substitutional sense of the word. Notably it is only the fourth
indemonstrable that straightforwardly gives rise to such a problem of reinterpre¬
tation, since the other logical connections exhibited in the earlier indemonstrable
schemata, viz. if ... then ... and not both ... and ... represent specifically
binary connections, at least for the Stoics, and at least so far as the evidence tells
us. Of these, only not both ... and .. . readily admits of generalization to the
n-ary case, and there is nothing in the sources to guide us in choosing between
the generalization to At most one of the following is true and the generalization
to Not all of the following are true , interpretations which again coincide only in
the two-term case. If we suppose that they took conjunction seriously as a logical
connection, perhaps the second is the more natural; for there is nothing to require
the translation of the initial xou as both except in the two-term case. Even here a
slightly dissimilar case would arise if we tried to construct the generalized schema.
For in the two-term case, the connective not both ... and ... coincides in sense
with the Sheffer stroke, which, since it is not an associative operation, cannot, in
the n-term case, be straightforwardly thought of quantificationally. The sentence:
« I (P I 7)
The Megarians and the Stoics
505
would mean something like:
Either all of a ,/?,7 are true, or a is false.
There would, however, remain the problem that the third indemonstrable schema:
Not both the first and the second; The first; Therefore, not the second
is insufficiently general in form to define the class of arguments which the general
account of conjunction would license.
Now it is unfortunately convenient to treat Stoic logic, however fragmentary and
indirect our understanding of it, as a product of the same general understanding
of the issues that we ourselves are able to bring to bear. In this frame of mind,
we are apt to see our scholarly task as one of rational reconstruction in the light
of that general understanding. In such a frame of mind, we might well agree with
Josiah Gould [1970, p. 83] that
it is clear in each of our fragments that the author intends the adjec¬
tive ‘undemonstrated’ to qualify what we would today call ‘argument
forms.’
and that the examples given are
what we would today call substitution instances. [Gould, 1974, p. 84]
Better to ask of the fragmentary information available to us, what stage the Stoics’
general understanding might have reached, allowing the relics of their doctrines a
reasonable degree of tentativeness without assuming that their approach, had it
only succeeded, would have been our own. This is, admittedly, a delicate task,
not least because we cannot know whether we have succeeded in it. But the
approach permits us, as need arises, to say ‘They did not foresee this difficulty’
rather than ‘This view creates a difficulty on the modern understanding and must
therefore not be attributed to them.’ As an exercise, one might ask whether, on
the evidence, the Stoics had hit upon something like our notion of logical form. If
they had, well and good, but if they had not, then we ought not to suppose that
all of the indemonstrables were regarded as formally valid or correct schemata in
any single sufficiently well-defined sense of ‘formal’ to be of use. We would not
be compelled, as we are by the contrary assumption,to assert of them that their
use of fjtoi ... fj ... in the fourth and fifth indemonstrables was a technical
one according to which it meant what is meant by 0110 disjunction. As we have
seen, unless they meant different things by rjxoi ... fj ... in different contexts,
0110 disjunction is not what they meant anyway, even if there is something, in
the relevant respects determinate, that they did mean. In spite of what we have
said about the notion of form, there is no harm in applying the word formal to
the Stoics’ work. By some standards, it is not informal and by those standards
we may therefore call it formal where that is the contrast intended; and we may
therefore distinguish their uses of fycoi and fj in formal contexts from their uses
of them in merely expository ones, where by this we mean just to distinguish the
ceremonial from the everyday.
506
Robert R. O’Toole and Raymond E. Jennings
10 THE LINGUISTIC EVIDENCE
Was there something that fjTOt and rj meant? What is the evidence? Quite apart
from the remarks of the early commentators, there is the evidence provided by
the Greek and Latin languages themselves. It is an urban myth that there is an
exclusive sense of or in English, and a suburban myth that Latin lexically marked
the distinction between 1110 and 0110 disjunctions by vel and aut. It is unclear
when these myths first arose. We have been unable to find them in any sources
earlier than the twentieth century. It is true that the Latin commentators used
aut ... aut ... to convey the Stoic use of rjxoi ... rj ... , but we must not place
too much weight upon this. It was the best choice on grounds quite separate from
the fictional one that aut corresponded to exclusive ‘or’. We should recall that
in the course of explaining the truth conditions of what he takes to be the Stoic
notion of Ste^Euypcvov, Gellius uses aut in a long disjunctive antecedent clause
of a conditional which is transparently intended to abbreviate a conjunction of
conditionals:
... si aut iiihil omnium verum aut omnia plurave quam unum vera
erunt, aut quae disjuncta sunt non pugnabunt, aut quae opposita eorum
sunt contraria inter sese no erunt, tunc id disjunctum mendacium est
... Noctes Atticae 16.8.14.
( ... if none of them is true, or all or more than one are true, or the
contrasted things do not conflict, or those opposed are not contrary,
then it is a false disjunction ... )
Evidently the choice of Latin vocabulary in which to cast the connective of the
fourth and fifth indemonstrables was not dictated by the need to convey exclusivity
formally. Had no Megarian or Stoic ever dreamt of the fourth indemonstrable, the
most suitable Latin translation of the fifth indemonstrable and for the regimenta¬
tion of ordinary language arguments of the corresponding grammatical cast, would
nevertheless have used aut. There is no reason to suppose that the mere use of
aut, independently of ancillary discussion and explanation of what it was intended
to convey, would have made the formal correctness of the fourth indemonstrable,
or of particular instances of it, transparent to Roman commentators.
Greek, like Latin, possessed no special connective by which 0110 disjunction was
distinguished from 1110 disjunction. The ‘logical’ sense of rjxoi ... rj ... and
its variants was essentially that of either ... or ...; like either ... or .. ., its
use was indifferent as to the number of terms joined and as between exclusive
and non-exclusive fillings; any additional imposition of an exclusive reading was
through emphasis and intonation. In particular the use of rjxoi as an auxiliary had
no special role as an indicator of exclusivity, that particle being a compound of rj
meaning variously or or than, and the enclitic tol an etymological cousin of the
second person singular pronoun. Its ordinary use was emphatic, akin to the use
in English of now surely or in Welsh English of Look you. Galen reports rj as an
alternative to rjxoi in Stoic usage, although he himself uses the latter exclusively
The Megarians and the Stoics
507
in the context of the indemonstrables [Frede, 1974, pp. 93-4], Certainly the use
of fjxoi ... fj... in ordinary non-philosophical written Greek was uncommon by
contrast with some philosophical writing and there is evidence that the philoso¬
phers have pressed into use a construction normally reserved as a spoken form.
Thus Denniston [1954, p. 553]:
fjxoi ... fj ... (often ryroi ... ye ... fj) is common in Plato and
Aristotle. It is difficult to say what degree of vividness toi retains
here. On the one hand, Thucydides confines fjxoi, like simple xoi, to
speeches ... this suggests that he felt xoi as vivid in the combination.
On the other, the frequency of fjxoi in the matter-of-fact style of Aris¬
totle suggests that for him xoi did nothing more than emphasize the
disjunction.
Bux ove ouyr|x vox xo ivcpep (ppop xrju; xqax fjxoi ... fj ... is more common
than fj ... fj ... in Aristotle and Plato, or that either of them set aside the
former for uses which prefigured the Stoic use. Neither is by any means true. In
particular, Aristotle uses fj ... fj ... in the overwhelming majority of cases,
and in many which would have provided excellent examples of disjunction for the
Stoics:
Ilp&xaaic; [iEv obv sxi Xoyoc; xaxacpaxixoc; fj axocpaxtxoc xivoc; xaxa xtvoc;.
ouxoc; 8s fj xcrOoXou fj ev |j£pei fj aSiopiaxoc (Prior Analytic 24“ 16).
(A premiss then is a sentence affirming or denying something of some¬
thing. This is either universal or particular or indefinite.)
auXXsXoyiaxat oxi aauppExpoc; fj auppsxpoc fj Siaytexpoc (46 6 31).
( ... he has proved that the diagonal is either commensurate or incom¬
mensurate)
and others where, if his understanding of the meaning of fjxoi ... fj ...
anticipated the Stoic use of it, we would expect fjxoi ... fj ... :
obiav yap ijfiiov Dvrycov fj Otcotcou fj cbrouv ectxi (46 6 15).
( ... every mortal animal is either footed or footless)
On the other hand, Aristotle’s uses of fjxoi ... fj ... , either give no evidence
that he was after a distinction that anticipated the Stoics’, or else provide evidence
that he had no such intention. Thus, when in the course of explaining kinds of
contrariety he denies that everybody must be black or white, he uses fjxoi ... fj
out) ya-p xa'v fjxoi Xeuxov fj peXav ectxiv (Categories 12“ 13).
( . . . not everyone is either white or black)
508
Robert R. O’Toole and Raymond E. Jennings
but the reason for denying this is that there are intermediates between white and
black, namely all the other colours. It is in any case used here between predicates
and not between whole sentences.
The relative scarcity of rj ... fj . .. as opposed to rjxoL ... fj ... in the
logical setting does not of course indicate that the Stoics gave the word fjiot any
special technical sense as distinct from a technical use. The thesis that it has a
special sense is forced upon us only if we also adopt the view that their enter¬
prise was a formal one in the substitutional sense. We have already seen that on
any straightforward interpretation, it was not. A more plausible guess would be
that that combination gradually gained favour in general philosophical practice
and presented itself to the philosophical innovator, perhaps Theophrastus, casting
about for suitably perspicuous notation as a construction already set apart for
special philosophical applications. Compare the current use of It is not the case
that .... Again, it need hardly be said that there was no special intensional sense
of fjxoi or fj or fjxoi ... fj ... in ordinary Greek, the necessity or contingency of
a disjunction being entirely determined by its disjuncts. But insofar as intended
exclusivity can be conveyed by emphasis, the intention is conveyed more easily
with more syllables than with fewer. And on that score fjxoi is more emphatic
than rj. Greek, like Latin and, come to that, like English, had a great variety of
connectives all of which could receive translation as or, but whose significance in
discourse is best understood by immersion in the literature in which they occur.
Like sive ... sive ... of Latin, Greek had die ... die which was common as
a conjunctively distributive connective in the antecedents of conditional construc¬
tions. Homeric Greek sometimes has xe in place of a second f), and rj xctt , in place
of a second die . Aeschylus sometimes answers die with dxe xai . But Greek
was in general more fluid in its use of particles than Latin. There are recorded
instances of fj ... xcd ... where fj ... fj ... would be expected; and there
are idioms in which xori occurs with the sense of rj, as in
av'Opcurcivr) aotpia oXiyou xivo<; odjla ectxiv xai oOSevot;
Human wisdom is worth little or nothing
ydeci xai xpwrjv
yesterday or the day before.
In general, the use of particular particles in the Greek of the last several centuries
of the old era varied, not only over time, but from author to author, even from
work to work, and particularly from genre to genre. 162 As a symptom of this
greater fluidity, there is evident a larger freedom in the use of particles in abbre-
viative constructions, especially favouring the use of constructions relying upon
superficial grammatical ellipsis over those requiring (or rather, as our thesis de¬
mands, at least capable of receiving) a truth-functional logical transformation. The
use of or constructions in (/-clauses to force a conjunctive reading is reinforced
162 For a detailed authoritative discussion, see Denniston [1954],
The Megarians and the Stoics
509
by the availability of a non-elliptical reading for an antecedent in and, particu¬
larly in a language which, like Latin, is less subtly variable than Greek in its use
of particles,. In Greek, for whatever reason, this tension between grammatical
ellipsis and logical transformation was less insistent than in Latin, so that when
the context demands it, an f/-clause occurrence of xch more readily accedes to a
conjunctively distributive reading. And one finds xch sometimes following eiTtcp
and grammatically absorbed by it, producing something akin to if even ... , as
one finds them in the opposite order having the sense of even if .... The logical
particles whose English counterparts we have been taught to think of as, to extend
Ryle’s colourful metaphor, importantly combat-ready, lived altogether more easily
in one another’s company.
Any attempt to construct a useful formal system that still retains a connection
with the inferential practices that have inspired it cannot but sacrifice non-logical
distinctions, and the logic of the Stoics, arising as it did out of a language so fluid
in its particulate usages as the Greek of their period, was not to be excepted. The
abstraction of the logically essential into a simplified vocabulary was part of the
task, but refining their very conception of the task and what was essential to it
was all a part of the same continuing academic enterprise. As modern logic has no
distinct notation for whether ... or ... , letting if... or ... do the work, and as
Roman logicians did not retain sive ... sive ... or turn ... turn, so Greek logicians
shed eix£ ... cite as they did the distinction between the suppositive negating
adverb pf| and the absolute oux (ouc , oux ) since the retention of pi) in negated if*
clauses would complicate conditionalization of arguments of the form of the second
indemonstrable. In any case, however much greater fluidity there may have been,
and however much simplified the account of logical connection, it remains true
that the role of xcu in Greek was preponderantly aggregative or agglomerative.
And the role of fjxot ... fj ... was preponderantly separative, as that of either
... or ... is in English. Its ordinary understanding was certainly such as to
support an inference schema such as the fifth indemonstrable. But the use of rjxoi
fj ... in the fourth indemonstrable goes beyond simplification. For, as we
have noted, there was, in Greek as in Latin and English, no or word that indicated
exclusive disjunction. If the Stoics intended that the fourth indemonstrable should
be understood formally in the substitutional sense, they could not have counted
upon that formal correctness being evident from the ordinary understanding of
its logical vocabulary, as they could have in the case of the first, second, third and
fifth. Consider, for example, a remark of Sextus:
16 8s 8ieC£UY[i£vov §v ex EL T “ v Ev auxco aAryOc, i ctv dpcpoxcpa fj
aArydfj fj dpcpoxepa (jjeu8fj, rpeOSoc eaxai xo oXov
... the disjunction has one of its clauses true, since if both are true or
both are false, the whole will be false. (SE AM 8.283)
Ei8evxAi}>, xr]E X aax oc;t;uppEv<;e 09 fj is not to be understood ‘in an exclusive sense’,
in spite of the exclusiveness of its disjuncts, but rather in the ordinary sense
which makes the conditional in whose antecedent it occurs elliptical for, or at least
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Robert R. O’Toole and Raymond E. Jennings
equivalent to, a conjunction of conditionals. The inequivalence of the conditional
having an exclusive disjunctive antecedent (in modern notation),
(a Y. ( 3 ) —> 7
and the corresponding conjunction of conditionals
(a -4 7) A (/? 7)
has consequences elsewhere. On a purely syntactic/semantic understanding of
Sie^euypsvov by which it would mean just declarative sentences joined by f) un¬
derstood in the technical 0110 sense, that inequivalence would make it difficult
to square the fondness for the dilemma, which was ubiquitous from pre-Socratic
through Hellenistic writings, with the general acceptance of the principle of con-
ditionalisation, which was generally accepted by the Stoics. Now the ordinary
application of conditionalisation as a test of validity would conjoin the premisses
in the (/-clause of a conditional, having the conclusion as then-clause. Presumably,
in the particular application, an argument involving dilemma would yield a pair
of conditionals whose validity would then be considered. The difficulty lies in the
fact that the conditional
(a Vfl. /?) >- 7
might be necessarily true because both a and /? were necessarily true and aV a f3
therefore necessarily false, but both conjuncts of
(a —> 7) A (/? —► 7)
false because 7 was false. On those grounds alone, it is unlikely that SicCeuypcvov
was a simple syntactic/semantic item in the Stoic conception of logic.
10.1 Cicero’s clanger
There is further evidence of this tension between the normal use of fjxot ... fj
and the Stoic use of it in the fact that there is a greater confusion sown in the
accounts of SieCeuypevov than there is in the accounts of the other connections.
If the fourth indemonstrable was intended as a formally admitted schema in the
substitutional sense, the difficulty can only have been one of understanding a new
technical sense being lent to the grammatical form rjxoi ... r) ... It cannot
be confidently rejected that Cicero, whose faux pas in his Topica still costs him
invitations, was among the victims of the confusion.
At least it must be said that a formal reading by which the fourth indemonstra¬
ble does represent a new technical use of fj ought to dispose us more charitably
toward Cicero’s curious augmentation, in top. 13.57 of the standard five indemon-
strables. Cicero claims there, so far as we know erroneously, that there was a Stoic
indemonstrable the Latin form of which would have been:
The Megarians and the Stoics
511
Non et hoc et illud; non hoc; Mud igitur.
Not both this and that; not this; therefore that.
When the indemonstrables are understood formally, this would seem on first hear¬
ing to represent a truly resounding logical clanger. Since the indemonstrables are
almost universally regarded as formal, this estimation has been the conventional
view. 163 There is no independent evidence that any Stoic logician ever included
this kind of argument in his list of indemonstrables. On this point, we take it that
Cicero was merely wrong. But could there have been such a kind of argument?
If we are right about what inferences could be justified by reference to the mean¬
ing of fjxot ... fj ... in Greek, and if the indemonstrables are formal, then the
use of fjxot ... fj ... in the fourth indemonstrable forces an exclusive reading
which did not exist in the natural language. For there, the nearest we have to an
exclusive fjxot ... fj ... is the use of fjxot ... fj ... with exclusive alterna¬
tives. But then the analogous technical use of not both ... and ... would force a
reading according to which from the falsity of one element the truth of the other
would follow. Indeed, anyone whose understanding of Stoic logic was indirect and
conjectural, and whose knowledge of Greek was not, might well have considered
that given the eccentric character of the fourth indemonstrable, the Stoics could
be expected to have a corresponding dual eccentricity of the sort embodied in
Cicero’s argument. It is true that the use of not both ... and ... never implies
by itself that both sentences cannot be false, but neither is there a use of or that
implies by itself that both sentences cannot be true. However, there are uses of not
both ... and ... with sentences which cannot both be false, just as there are uses
of or with sentences which cannot both be true. Understood as a formal theory,
there is nothing more eccentric about Cicero’s supplement than there is about the
undoubtedly Stoic fourth indemonstrable. But suppose for the sake of argument
that Cicero’s addition were to be found extensively in Stoic logical writings and
attributed, say, to Chrysippus. Any historian of Stoic logic finding himself unwill¬
ing to accept that indemonstrable as merely representing a technical usage, ought
to feel no more willingness to accept, on those terms, the Stoics’ eccentric use of
‘or’ in the fourth.
Make the parallel more explicit. The formalist historian claims that the Stoics
used fjxot ... fj ... technically to mean Either ... or ... and not both ... and
... In ordinary Greek, its meaning comprehended the former conjunct but not the
latter. If Cicero were right, there would be a second pill to swallow: that the Stoics
used Not both A and B technically to mean the same thing. In ordinary Greek,
its meaning comprehended the latter but not the former. Even on a formalist
rendering, the mistake ought to seem on reflection no great logical howler. But
when we consider, as we shall, the notion that the indemonstrables were not for¬
mally intended, we may also entertain among others, the possibility that Cicero’s
163 The Kneales are a notable exception, and offer a plausible and detailed alternative account.
Bochenski has remarked (in conversation) that to ask Cicero about logic is about as sensible as
to enquire of Sartre about the writings of Carnap. Calvin Normore has offered that the error
may be imputed to Cicero’s well known insomnia. Both may well have some bearing.
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error represents at worst a merely historical error or a badly worded description,
a wrong but not unreasonable reconstruction from memory of something he had
read or heard from Diodotus or Philo. 164 But again, on a non-formalist construal,
especially given our more less comprehensive ignorance of the teachings of minor
Stoic teachers, it could well be an accurate recollection of something taught him
(however erroneously) by Diodotus. It would not have been an impossible kind
of argument, on a non-formalist view, for a Stoic to have noted. Consider what
textbooks of this age say about aut.
10.2 The question of form
So we return to the question whether the Stoics regarded the indemonstrables as
formally correct schemata in anything like the modern understanding of formal
correctness. The evidence is clouded and there are many imponderables. We do
not know with certainty to what extent the technical vocabulary, OyLife , validus,
SiE^EuypEvov and so on had been freed from its etymological roots for Galen or
Sextus, or with certainty what points of terminology and doctrine remained a mat¬
ter of controversy into the Christian era. We do not know with what exactitude
the logical vocabulary was defined by Chrysippus or others. But it would not be
too pessimistic at least to lower our estimations of their capacity for logical de¬
scription. As we have seen, the standard substitutional notion of valid form does
not adequately account for the Stoic account of disjunction in inference, since it
does not accommodate connectives of no fixed arity. A relaxed, descriptive notion
of valid form might come closer to theirs. The difference is this: a substitution
account presents a schema and (perhaps implicitly) a rule of uniform substitution,
or asserts that for every pair of sentences a and b, such and such a conclusion may
be inferred from such and such premisses. Arguments of the same form retain
the syncategorematic vocabulary and repeat sentences in the same pattern as the
repetition of metalogical variables in the schema. One might say that the sub¬
stitutional account stands for an abstract syntactic description applicable to any
argument of the form. What we shall call a descriptive account would give an ex¬
plicit description, saying what belongs in each premiss, and what in the conclusion,
perhaps illustrating by a schema, or an example. Of the fourth indemonstrable it
might say: ‘An argument of the fourth type has a diezeugmenonic major premiss
etc.’ and mention that a diezeugmenonic sentence is of the grammatical form:
fjxoi to a fj to /3
(not ‘a diezeugmenonic sentence is any sentence of the grammatical form
f]TOl to a f] to /?’.)
The class of valid arguments of that descriptive form would be the class of ar¬
guments satisfying the description, which might but might not coincide with the
164 Particularly bearing in mind that the account in the Topica is a reconstruction written, not
in a library, but during a journey.
The Megarians and the Stoics
513
class of arguments obtained by uniform substitution in the illustrating schema. In
the case of arguments with a diezeugmenonic major premiss, presumably the two
notions would not coincide if the understood arity of disjunction were variable.
If the distinction between the two notions of form were never explicitly stated, it
is credible that discussions would sometimes vaguely have assumed the one and
sometimes vaguely the other.
In the case of the fourth indemonstrable, a substitution account would offer the
schema:
Either a or /?; but a; therefore, not /?.
A rule of substitution would license any argument obtained by substituting an
occurrence of some declarative sentence A for every occurrence of a and an occur¬
rence of some declarative sentence B for every occurrence of /? as an argument of
the form of the schema. Alternatively, a substitution account would say:
For every sentence a and every sentence /?, from a or /?, and a, not-/?
may be inferred.
A descriptive account of [IV] (for the general case) would be this: ‘From a dis¬
junction together with one of its disjuncts, the negation of any distinct disjunct
may be inferred.’ What constitutes an argument of this description depends upon
what is meant by disjunction, but we may say that the simplest argument of this
kind would of be the form:
Either the first or the second; the first; therefore, not the second.
Now if, in addition, our notion of disjunction had as its foundation the notion of
a relationship between states of affairs or situations such that exactly one of them
must obtain (and only derivatively of a string of sentences alleging such states,
separated by or), rather than simply any string of sentences separated by or, the
puzzle about the technical meaning of fjxoi ... rj ... would be less perplexing.
Indeed there would be no puzzle. Both the fourth and fifth indemonstrables would
represent valid kinds of argument, and the schemata presented would represent the
forms of the simplest arguments of this kind. Why Either ... or ... ? It is the
obvious connective, since it permits the construction of a true sentence out of
contradictories and, in any case, the Either ... or ... construction is the one in
which these contradictory alternatives are naturally contemplated. That the fifth
indemonstrable is justifiable solely on the basis of the meaning of or and the fourth
only on the basis of the particularities of its arguments, on such an account, does
not matter. It is relevant only in the presence of convincing evidence that they
had in mind a substitutional notion of form. That is precisely what is lacking.
The evidence suggesting the less finely tuned notion of validity is by no means
unequivocal. The clearest case of a descriptive presentation of the indemonstrables
is that of Ioannes Philoponous in his Scholia to Ammonius:
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Robert R, O’Toole and Raymond E. Jennings
The disjunctive syllogism proceeds on the basis of complete incompat¬
ibles. 165
But all of the early sources give, more or less, a descriptive account of the fourth
and fifth indemonstrables. Cicero, who gives barely more than schemata, feels
it necessary to add the comment that ‘these conclusions are valid because in a
disjunctive statement not more than one (disjunct) can be true’, a remark more
significant for having seemed necessary than for what it says. Much that is oth¬
erwise puzzling is less so on the view that their notion of validity had, at least
not yet, become fixed upon a substitution account. If the notion of disjunction
was the descriptive one, meaning essentially sentences in a certain relation, every
disjunction which was uytrjc; or validus in the more etymological sense of ‘proper’
or ‘sound’ would also be uyuy; or validus in the derivative sense of true, even in
the further derivative sense of valid. This would explain Gellius’ rejection of the
premiss
Aut honesta sunt, quae imperat pater, aut turpia
A father’s commands are either honourable or base
on the grounds that it is not what the Greeks call ‘a sound and regular disjunction’
(uyifjc; et vopipov 8ie£euyp£vov). (Gellius Nodes Atticae 2.7.21) It would justify
Favorinus’ response to Bias’ dilemma, that its major premiss (You will marry either
a beautiful or an ugly woman) was not a proper disjunction ( iustum disiunctivum ),
since it was not inevitable that one of the two opposites be true, which must be
the case in a disjunctive proposition. ( Nodes Atticae 5.11.8) On the substitution
account, the truth or falsity of a premiss ought not to affect validity. On the
descriptive account, particularly in the case of a disjunctive premiss, its falsity
cannot but affect at least the question whether it is of the particular valid kind,
since if it is false, it is not a genuine disjunction. There are other, similar instances,
as for example Sextus’ rejection of the argument:
Wealth is either good or bad; but wealth is not bad; therefore, wealth
is good
on the grounds that the first premiss does not state an exhaustive disjunction of the
possibilities. 166 . These‘extra-logical considerations’[Gould, 1974, pp. 165-66] and
this ‘serious confusion between a disjunction and a true disjunction’ [Mates, 1953,
pp. 52-53] have puzzled earlier modern commentators. But if the specification of
form was thought of as being given descriptively rather than substitutionally, so
that the distinction between disjunction and true disjunction was not present, then
the inexhaustiveness of the major premiss would debar justification by reference
to [V] as the incompleteness of conflict would debar justification by reference to
[IV]. And notice the restricted claim Sextus Empiricus is, on one occasion, content
to allow himself about the nature of disjunction:
165 Ammonius in an. pr. Praefatio xi. The translation is Mates’s [1953, p. 131].
166 SE AM 8.434
The Megarians and the Stoics
515
to yap uyt.ec SLE^EuypEvov ExayyEXXETaL sv xuv ev auxto uyisc; elvat, to
8e Xoltiov rj xa Xoim i^eOSoc r) (JjEuSfj pcxa paxr|c; ( SE PH 8.191).
The true disjunction declares that one of its clauses is true, but the
other or others false or false and contradictory.
It is a curious restriction if the distinction between a disjunction and a true dis¬
junction is an important one.
Again, Galen’s discussion of the distinction between SiE^EuypEvov and rapaSiECeuypevov
makes it clear that he at least does not understand the claims of the fourth and fifth
indemonstrables according to a substitution sense of validity. For he recognised
what could be called paradisjunctive syllogism as a distinct type of syllogism, while
evidently not regarding it as exhibiting a distinct form. Having given an account
of Chrysippus’ classification of the indemonstrables, he remarks:
In syllogisms of this sort be., disjunctive and hypothetical, the major
premisses determine the minor; for neither in the disjunctive do more
than two additional premisses occur nor in the conditional, while in
the case of incomplete conflict (eXXoit) paxfl) it is possible to make one
additional assumption only. ( Institutio Logica 7.1)
But when earlier he discusses the distinction between complete and incomplete
conflict, a single multi-termed sentence does duty for both.
For ‘Dion is walking’ is one simple proposition, and also ‘Dion is sit¬
ting’; and ‘Dion is lying down’ is one proposition, and so, too, ‘He is
running,’ and ‘He is standing still,’ but out of all of them is made a
disjunctive proposition, as follows: ‘Dion is either walking or is sitting
or is lying down or is running or is standing still’; whenever a propo¬
sition is formed in this way any one member is in incomplete conflict
with another, but taken all together they are in complete conflict with
one another, since it is necessary that one of them must be true and
the others not. ( Institutio Logica 5.2)
Notice that as an example of conflict, Galen’s is a good one in its listing states
that cannot simultaneously obtain, but a bad one in providing a list that is not,
as Galen suggests, genuinely exhaustive. (Dion might be crouching or signalling.)
From an inferential point of view, its inexhaustiveness is unimportant, since given
the truth of the disjunction it follows, solely in consequence of the meaning of the
particle rj , that if one of the disjuncts is false, then one of the others is true.
Ian Mueller, in his discussion of the possible non-truth-functional status of the
Stoic sentential combinations says
We cannot be sure about ‘or,’ but I suspect that a disjunction was taken
to be true only if the disjuncts were mutually exclusive and exhaustive
of the alternatives. [Mueller, 1978, p. 20]
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Robert R. O’Toole and Raymond E. Jennings
It is a more plausible conjecture that what was meant by ‘disjunction’ was what
would be called ‘true disjunction’ on a substitution interpretation, that in the case
of disjunction, the etymological sense of ‘sound’ suggesting the correct internal
relationships among parts, was not absent from the understanding of uyif)t; ■ In
an application to a very simple object, this sense would be tantamount to ‘true’
in the sense of ‘genuine’. (For in a sufficiently simple kind of object, little in the
way of internal relationships can fail before the object is not merely defective of
the kind, but no longer an instance of the kind.) No disjunction that was true in
the sense of ‘genuine’ could fail to be true in the sense of ‘representative of how
things are’. Now to say this is not to say that they were confused between form and
content as we imagine we understand the distinction. It is to say that the boundary
between the two had not yet been clearly drawn, let alone drawn where, at least in
propositional logic, we now draw it. In fact, we can go further. For one need only
read De Morgan’s Cambridge lectures to see how far much of the philosophical
establishment as recently as the nineteenth century was from grasping our present
understanding of form. The resistance from Sir William Hamilton and his followers
to the liberation of the idea of logical form from the shackles of Kantianism was
one of the most serious academic obstacles that De Morgan had to surmount in
getting his logical ideas accepted. One could argue that that struggle for liberation
was one of the major contributions of nineteenth century logic. And the battle
will not be certifiably won until the day when logic textbooks no longer call 1110
disjunctions exclusive on the grounds that their disjuncts are incompatible. It is
therefore a serious matter to suppose that the Stoics were in full possession of
the notion, particularly when the historical evidence indicates so clearly that their
conception of the nature and place of logic were fundamentally different from that
of twentieth century theorists.
Finally, when the matter is viewed in this light, one is tempted to speculate
that it was precisely their preoccupation with the dilemma both as a form of ar¬
gument and as a paradigm of moral predicament which fixed the attention of the
later Peripatetics and the Stoics upon inference patterns such as [IV], as having a
fundamental place in a codification of academic inferential practice. If this were
true it would not be surprising that they would wish to exclude as improper those
disjunctions of which both disjuncts could be true, for these are the just the in¬
stances which would defeat conditionalisation of dilemmas. Since the substitution
account of validity would not rule out such disjunctions and a descriptive account
would, the descriptive account seems on that score to be the more likely candidate
for the Stoic conception of validity.
11 WHAT STOIC DISJUNCTION MAY HAVE BEEN
We have concentrated upon disarming the prejudices which might suggest that
they introduced a technical sense of rjxoi... rj ... and aut ... aut ... or relied
upon senses of those constructions already present in Greek and Latin. But we
have allowed, without comment, anachronistic terminology such as ‘proposition’
The Megarians and the Stoics
517
to creep into our account (albeit only descriptively) which might itself create the
false impression that the Stoics had some such notion in common with us. It
would in particular be a serious misunderstanding to suppose that the Stoics had
in their notion of a lekton a notion corresponding to the Fregean proposition. It
is precisely in the nature of the lekton that most recent commentators have found
grounds for denying the earlier assumption that the nature of Stoic logic could
be well enough understood by comparison with modern calculi. Their arguments,
to which may be added the arguments given here against the application to their
work of a substitutional notion of logical form, have drawn their premisses from
quite a different source, namely the nature of the relationship between Stoic logic
and Stoic epistemology and physics.
One account which forcefully presents Stoic logic in a non-formalist interpre¬
tation, is due to Claude Imbert [1980]. She takes as her point of departure the
Stoic notion of cpavxaata , usually translated as ‘presentation’, taken up as an al¬
ternative to Aristotle’s theory of imitation and applied to the art of Alexandria.
It is through an understanding of the nature of (paviaoiai and their relationship to
the major premisses of the indemonstrables that we understand why Stoic logic is
conceptually incomparable with modern calculi.
The conclusion of a Stoic syllogism is inferred from other sentences
which translate natural signs apprehended in presentations, and which
never presuppose the existence of transcendent forms or universals
... Every logical structure rests on the possibility of translating pre¬
sentations into discursive sequences, and each sequence must exhaust
the scientific content latent in its presentation. Inference thus depends
on a rhetorical function which maps utterances ( lekta ) on to contents
of presentations (phantasia). [Imbert, 1980, pp. 187-88]
The transition from impression, which all animals have, to a presentation charac¬
teristic of human apprehension, depends upon the capacity to grasp connections
among the contents of experience. Complex utterances, hypothetical, conjunctive
and disjunctive, represent three ways of grasping connections. The one which
concerns us here is the way which corresponds to the disjunctive proposition: the
recognition of alternative exclusive possibilities. The use of the language of cpav¬
xaata in this connection is suggestive, in one respect, of Aristotle’s use of the same
term in De Anima where it designates an activity characteristic of common (as
distinct from particular ) sense. And other evidence has suggested to some com¬
mentators that the ideas of the logical connections were originally a Peripatetic
innovation. 187 Finally, a full understanding of the Stoic preoccupation with what
appears to the present day philosopher as a rather specialized and arcane notion
of disjunction cannot neglect its connection with a theme which recurs as a leit¬
motif in one form or another throughout the history of Greek philosophy. The
167 See [Barnes, 1985] for a discussion of the evidence suggesting that Theophrastus was one
Peripatetic source.
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Robert R. O’Toole and Raymond E. Jennings
SisCcuYpevov of the Stoics is a late practical refinement of the notion of the con¬
flict of opposites, which can be traced through Heracleitus’ doctrines of the unity
of opposites to Anaximander’s doctrine of the generation of opposites from the
undifferentiated cotetpov , and is to be found in the central images of the mythic
cosmogony of Hesiod. For the Stoics, it was at the heart of their ethics, physics
and logic, and its recognition was a necessary constituent of the rational unity to
be made of the conduct of human affairs and the operations of nature. We can
construct a simplified model that realizes some such conception as the one they
seem to have had in mind. According to such a picture, each succeeding state of
the world makes some atomic sentences true and the rest false. So each moment of
time may be thought of as a function or rule which takes sentences to truth values.
Coming to an understanding of the intelligent character of the world amounts to
grasping the principles by which these functions are selected in their turn. And in
a poetic or spiritual frame of mind, we might imagine such rules as competing for
selection and thus, since they represent incompatible assignments of truth-values,
we might imagine them as being in conflict. Moreover, the image comes equally
to mind of nature selecting its way among these competing functions according
to some rational principle. In the sphere of individual action, the notion will
readily suggest itself to us that in minute part we each bear some responsibility
through our choices for the successive states of the universe. The apprehension
of the distinctness of these state-functions within the subdomain of alternatives
presented to us would, in this admittedly fanciful reconstruction, correspond to
the apprehension of SteCeuypevov . Like the rows of a truth table, or the items of
a menu, they would be represented as mutually exclusive alternatives; were we to
articulate them, it would most naturally take the form of a string of alternatives
separated by ‘or’: this set of atoms true or that or the other ....
Now this fancy is an anachronism, though the Stoics seemed to recognize some¬
thing like the possibilities represented by the rows of a truth table. But if we
cannot understand the Stoic use of or in other terms than those of twentieth cen¬
tury logical theory, it would be less misleading to bring the notion of SieCeuyfievov
into the light of such simple-hearted model-theoretic ideas than to associate it
with the substitutional idea of a particular logical form.
As much as one might wish to complain to the Stoics that there are connections,
such as non-exclusive alternatives, which are not provided for in their scheme, such
objections are not to the present point, for what we have wanted is an explana¬
tion of the Stoic use of fjxoi ... rj ... which accords with the evident fact that
their technical use does not constitute a technical meaning. Understood as rep¬
resenting the most succinct way in which we reflect in utterance the connection
between exclusive alternatives viewed as such, the use is surely unobjectionable.
The fact that we, and for that matter, the Greeks, had other less succinct ways of
reflecting such connections and as well used the same connective for non-exclusive
alternatives is neither here nor there. In any case, when such alternatives confront
us, a complete analysis of the possibilities will always yield exclusive alternands,
namely, those corresponding to the three l’s of the truth table of V. If a and /?
The Megarians and the Stoics
519
present themselves to us as non-exclusive alternatives, our choice, when fully and
analytically apprehended, is seen to be among the three exclusive alternatives:
pursuing both a and /?, pursuing a but not /?, and pursuing /? but not a. Though
the origins were different and the motives, the method need not be thought en¬
tirely unlike Boole’s. For he too took exclusivity, even the same arity-free idea
of exclusivity, to be centrally important to his representation, but the exclusivity
was constructed out of a non-exclusive disjunctive use of or.
12 STOIC DISJUNCTION AS A HYPER-RELATION
It is important to bear in mind that before the nineteenth century, logical the¬
orists, though they spoke of form (as distinct from content) thought of logical
connection in relational rather than formal terms. The character and status of
the items between which the relations were thought to obtain varied through the
history of the subject, but the relational character can be said to have persisted
without challenge at least until George Boole’s temporal semantics for the con¬
nectives, and in some branches of logic, notably Idealist logic, to have persisted as
explicit doctrine well into the twentieth century. The 1929 symposium on negation
(Mabbott et al. 1929) might be said to mark its final departure from academic
philosophical logic. We can, however, capture the character of Stoic disjunction
in the recent language of coherence measures.
Let E be a set of sentences. Then the coherence level of E, A(E) = min£ : 37 r G
n € (S) : Vc € 7 r,cFl, if that limit exists; else A(S) = oo.
Thus, for example, A({p, -ip}) = 2; A({p A q,p A -> q, ->p A q, ->p A ~^q}) = 4; A({J_
}) = oo and so on.
Let E be a set of sentences. Then the coherence dilution of E,A(E) = min£ :
3A C E |A| = £, and A FT, if such a subset exists; else <5(E) — oo.
Thus, for example, 5({p A ->p} = <5({Y} = 1; <5({p A q,p A ~>q, -ip A q, ->p A ~ i q}) =
2; 5({a,a —> /!,/? -4 7,7 -4 £,->£}) = 5; <5({p}) = 00 and so on.
Again, maximum dilution is illustrated by the modest believer, whose only
mistaken belief is that at least one of his beliefs is false. No proper subset of his
beliefs is inconsistent, yet the set as a whole is.
Then a set-representation of an n-term Stoic disjunction can be given as the set
E = { 07 ,... ,cr n } of its disjunctions where E satisfies:
A(E) - 6 HE]) = |E| = n.
The weaker Stoic notion of paradisjunction can be given a set-representation
that weakens the dilution requirement to
520
Robert R. O’Toole and Raymond E. Jennings
«HS]) ^ |E|.
Now it would not have taken the Stoics beyond the resources available to them to
have introduced a measure on paradisjunctions representing the maximum number
of disjuncts that could be true. Such a measure would indirectly have yielded a
measure of dilution of incoherence capable of independent study. As an example,
consider the set
E = {P -t Q,Q -tfbPAg}
A(E) = 1;
*HE]) = 2-
E is a set representative of a paradisjunction: one of its elements must be true
but all of them can be. On the other hand, the conflict among the elements of
—i[E] is less diffuse than among the negated disjuncts of a Stoic disjunction. If
the ideal is the absence of conflict, evidently more dilute conflict is better than
less. Thus the notion that inference should preserve dilution is in the logical spirit
of Stoicism. We conclude with the observation that a system of inference that
(a) permitted only dilution-preserving inferences, and (b) took those inferences
E a as correct for which the dilution of E U {a} was greater than the dilution
of E U {-ia} would satisfy connexivist constraints on inference corresponding to
the theses of Aristotle and Boethius discussed earlier (page 482). Such a system
was nearly within reach of the Stoics, and would constitute a natural extension of
their logical theory.
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vol. 3, 1879.
ARABIC LOGIC
Tony Street
INTRODUCTORY COMMENTS
This chapter limits itself to logical writings in the peripatetic tradition produced in Arabic
between 750 and 1350. It is intended to provide a tentative framework for analysis of
these logical works by describing aspects of the historical and intellectual context within
which they were written. This is done by testing the model put forward in Rescher’s
Development of Arabic Logic against accounts of the syllogistic in a number of authors.
By about 900, the Organon had been translated into Arabic, and was subject to in¬
tensive study. We have texts from that time which come in particular from the Baghdad
school of philosophy, a school which at its best proceeded by close textual analysis of the
Aristotelian corpus. The school’s most famous logician was Alfarabi (d. 950), who wrote
a number of introductory treatises on logic as well as commentaries on the books of the
Organon.
Within fifty years of Alfarabi’s death, another logical tradition had crystallized, finding
its most influential statement in the writings of Avicenna (d. 1037). Although Avicenna
revered Alfarabi as a philosophical predecessor second only to Aristotle, his syllogistic
system differed from Alfarabi’s on two major structural points. It is in consequence rel¬
atively straightforward to assign subsequent logicians to one or other tradition. Avicenna
differed from Alfarabi in his approach to the Aristotelian text, and assumed even less than
Alfarabi had that it contained a straightforward exposition of a coherent system merely
awaiting sympathetic interpretation to become clear. Due perhaps to the flexibility of the
larger philosophical framework with which it was associated, a framework which proved
adaptable to the needs of Islamic philosophical theology, Avicenna’s logic came in time
to be the dominant system against which later logicians set forward their own systems as
alternatives or modifications.
The success and rapid spread of Avicenna’s philosophy and logic elicited a strong re¬
action from establishment theology, whose very intellectual vitality was perceived to be
threatened. The clearest and most influential response to Avicenna was given about half a
century after his death by Abfl-Hamid al-GazalT (d. 1111). A case had been made at least
as early as Alfarabi that logic could help Muslim scholars in juristic and theological rea¬
soning. GazalT accepted these arguments and went so far as to preface his juridical summa.
The distillation of the principles of jurisprudence, with a short treatise on logic. Logic
continued to face pious opposition after GazalT, but even scholars who were opposed to
Greek philosophy in its various manifestations were agreed that, taken as a formal system,
logic was unobjectionable.
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
© 2004 Elsevier BV. All rights reserved.
524
Tony Street
Logic after GazalT was regularly studied by Muslim scholars for use in theology and
jurisprudence. It also continued to be studied by Muslim scholars who were interested
in the deeper formal and philosophical questions Avicenna had raised. GazalT extended
the number of students of logic by inspiring people to study basic logic as a kind of exer¬
cise in critical thinking, but had no impact on the vitality or nature of the existing logical
tradition. It would seem that the most interesting discussions in Arabic on logic were con¬
ducted from the late eleventh century on. Scholarly studies particularly in Spain carried
on in the tradition of the Baghdad school, culminating in the work of Averroes (d. 1198).
The Averroist project to make literal and globally consistent sense of the Aristotelian texts
was an extension of the methods of the Baghdad school, even though Averroes saw many
flaws in the work of his predecessors. Elsewhere, however, Aristotle had ceased to fig¬
ure as a major coordinate to which logicians referred in constructing their systems. This
role was rather filled by Avicenna’s system, which was modified, extended, and in part
rejected.
By 1350, it is clear that the Avicennan tradition predominated over the Farabian and
Averroist traditions throughout the Islamic world, and the systematic problems in Avi¬
cenna’s formal syllogistic were taken as being settled. By this time, texts were being
produced which continued to figure in the syllabus of the madrasa down until recent
times. For centuries after, advanced logical investigations continued in the Islamic world,
but the madrasa texts were always the way that Muslim scholars had come to be able to
conduct those investigations.
A note on conventions
All dates are given in common era, except occasionally in the bibliography. I have denied
diacritical machinery to all names of dynasties and places, and also to the scholars I
refer to by names that are either simplified, or derived from the medieval Latin tradition
(among others, Alfarabi, Avicenna and Averroes). All other names are given on their first
occurrence within a section in sufficient fullness to identify the scholar in question, and
afterwards in a shortened form; so, for example, FahraddTn ar-RazT becomes after first
reference to him merely RazT. Due to the vagaries of my grasp of BlBTgX, all names are
given in the bibliography without their final definite article.
In the translations, I have tended to standardize the names of Muslim scholars. Book
titles are given in translation in the text, and in Arabic in the bibliography. Many of the
texts presented here in translation have been translated before. When I refer only to a
translated version of the text, or to the translated version before the Arabic original, I
have followed that translated version verbatim. When I refer to the Arabic original before
the translated version of a text, I have relied on the translated version, but departed from
it in some way, even if only slightly. At those times that I have emended an Arabic text as
I have translated it, I mark the point that I have modified it with an asterisk.
Two frequent intrusions in the chapter may prove annoying. The first is constant cross-
referencing within the text—I hope this makes what is essentially a narrative somewhat
more chapter-like. The second is the phrase ‘at least on my reading’. The phrase is
intended to be disarming. A number of the texts used here are only in manuscript, or
Arabic Logic
525
are part of a larger opus demanding extended treatment, and my interpretation of them is
tentative.
I have instituted one semi-technical convention. When I refer to a logician as ‘Avi-
cennan’, I mean that he has put forward a system including the three elements I identify
on page 553 below. By contrast, when I refer to a logician as ‘post-Avicennan’, I just
mean that he lived after Avicenna had died. By this convention, all Avicennans are post-
Avicennan, but the reverse is not the case.
Two last points regarding conventions. Although I use ‘Spain’ to refer to what used
to be called al-Andalus, I refer to the logicians working there as Andalusian. The Index
gives occurrences of ancient and medieval logicians named in the chapter, and is intended
to serve as a point of reference for a set of names which for the Anglophone can be fairly
forgettable. Definite articles, even medial ones, are ignored in ordering index entries.
1 LIMITS, METHODS AND SOURCES FOR THE CHAPTER
In this chapter, I present a historical sketch of logical writing in Arabic. A number of
works have dealt with the broad topic of Arabic logic in the recent past [Arnaldez, I960-;
Black, 1998; Gutas, 1993; Inati, 1996; Madkour, 1969 2 ; Rescher, 1963c; ?; Rescher,
1967a], and though all have given at least a rapid historical outline of the subject, it seems
to me still to invite more extended treatment. This is so above all because understanding
the particular logical tradition within and against which a given logician writes determines
absolutely our ability to go on to appreciate and assess the nature and quality of the work
presented. When the output of a logician writing in Arabic seems incommensurable with
the work of a contemporary logician writing in Latin, it is nearly always as a result of the
different configurations of their respective logical traditions. In light of this, I have tried to
pull the existing secondary literature together to make clear the delineation of logical tra¬
ditions in the Islamic world. This has led me in sections 2, 3.1, 3.2, 3.3 and 4 to reparade
material which appears as such (in more or less extended form) in earlier writers, whereas
in sections 3.4, 3.5, 3.6 and 5,1 deploy material in ways less frequently encountered. The
upshot of all this is that I end up offering, with somewhat more detail of technical aspects
and traditional affiliation, the account first given by Ibn-Haldun (translated at page 580
below). In the remainder of this section, I try to justify the limits, methods and sources I
have used in writing the chapter.
Limits ‘Arabic logic’ is in four respects imprecise as a title for this chapter. Firstly,
because the logical works studied here consist of those written between 750 and 1350 (and
I concentrate only on those written between 900 and 1300), many Arabic logical works are
left to one side; these include a number of modern works contributing directly to the post-
Fregean logical enterprise. Secondly, many of the scholars studied here were Muslim.
They contributed to a tradition of writing which was made possible in the last analysis by
the Islamic conquests, a tradition which was carried forward in both Arabic and Persian.
In light of considerations like these, some would argue that a title like ‘Islamic logic’
would be more appropriate. Thirdly, although it is clear that Stoic logic filtered through
526
Tony Street
to scholars working in Islamic law and theology, there is no tradition of translating Stoic
works and commenting on them comparable with that devoted to Peripatetic works, and,
except for brief mention at page 556 below, I have left them out of consideration. Lastly,
for reasons I explain below in Sources, I confine myself to aspects of the syllogistic. A
more precise though less attractive title for this chapter, given its restrictions, would be
‘Peripatetic logical writings in Arabic produced in the realms of Islam between 750 and
1350, with special reference to the syllogistic.’
The study of medieval Arabic logic is not yet in a state remotely comparable with the
cognate study of medieval Western logic. The vast majority of the relevant texts are still
in manuscript, and a fair number of those that are available in print have not been edited
adequately. Important preliminary studies have been carried out, but the sum of these is
still a long way short of desiderata set down as long ago as 1965 [Mahdi, 1965]. This
state of affairs has two consequences for the writer of handbook entries. The first is that,
at best, only a sketchy and often conjectural outline of the history of the subject can be
given. The second consequence, more philosophically disappointing, is that although we
can point to various aspects of Arabic logical writings that are of philosophical interest,
we are not in a position to say that a given topic or set of topics as treated by logicians
writing in Arabic is more interesting or original than others.
Having noted the limits I have imposed on myself, or had imposed upon me by the
state of the field and the reach of my competence, I should go on immediately to dispel the
impression that I stop at 1350 because it is the end of original logical writing in Arabic, as
it is sometimes said to be. I am perfectly prepared to entertain the possibility that logical
production went through a radical decline in quality at this time. But it cannot simply be
assumed to be the case because the preferred genre of logical composition came to be the
commentary, or because (as one writer on Arabic logic put it): “Toute evolution sociale
monte et descend, progresse et tombe en decadence” [Madkour, 1969 2 , page 240]. We
simply have to read these texts. Sadly, I cannot claim to have done so—my knowledge of
Arabic logical texts written after 1350 is even sketchier than my knowledge of the texts
written before 1350.
Still, plausible reasons can be given for stopping at 1350. By that time, it is clear
that even in Spain, as well as in North Africa and Egypt, a system of logic which had
descended from the Avicennan tradition had come to predominate. By that time, texts
had been composed which continued to be commonly taught in the madrasa down until
recent years. Further, by this time QutbaddTn ar-RazT at-Tahtan! (d. 1365) had written
his book purporting to settle conflicts between two philosophical traditions in Persia and
Transoxiana—his contribution has been claimed to mark the end of a significant period in
the history of Arabic logic [Rescher, 1964, page 81]. Finally, 1350 is sufficiently recent
to include Ibn-Taymiyya (d. 1328), a great and very quotable hater of logic.
Aside from these limitations, this chapter is confined and configured by my own prej¬
udices. I think that disproportionate scholarly effort has gone to the study of the Baghdad
school at the expense of post-Avicennan logic. In consequence, I have dwelt rather more
on the logicians of Persia and Transoxiana in the thirteenth and fourteenth centuries than
most historians of Arabic logic do. I also say rather less about Averroes than most his¬
torians do. Western medievalists have tended to be more interested in the logicians and
Arabic Logic
527
philosophers of the Islamic world who were translated into Latin, such as Averroes. While
there is nothing wrong with studying Averroes and in recognising more fully the extent
of his contribution to Western logic, there is a danger in presenting a distorted picture of
the relative range, intensity and quality of logical studies throughout the Islamic world.
There is simply no doubt that the time and place of major output was in the east, in Persia,
Khurasan and Transoxiana, in the twelfth and thirteenth centuries—it may well also be
the time and place that most of the interesting insights and doctrines were formulated.
Methods Although a number of attempts have been made in the past to sketch an outline
of the history of Arabic logic, two in particular exemplify how widely approaches to
the task have differed. The first approach assumes that logic is somehow separate from
other philosophical doctrines, and is not fitted out to serve given metaphysical purposes.
Madkour writes of the logicians writing in Arabic that
II serait fastidieux de suivre ces logiciens dans leurs divers exposes; d’ailleurs,
il n’y aurait pas grand interet a mettre un tel projet a execution; car si les
philosophes musulmans different entre eux en ce qui concerne certains problemes
physiques ou metaphysiques, ils sont tous d’accord sur les grandes questions
logiques. [Madkour, 1969 2 , page 9]
This means that it only remains to find a paradigm author to give the systematic outline
of Arabic logical doctrine. Madkour would have preferred if that author could have been
Alfarabi, but given the fragmentary nature of his surviving writings, it has to be Avicenna
...qui represente a juste titre l’ecole arabe et offre une doctrine complete
sur laquelle on peut se prononcer aisement. Ses ecrits, que nous avons en
main, presentent les differentes manieres dont les philosophes musulmans
ont traite la logique aristotelicienne; ils en contiennent des abreges tres precis
et des commentaires assez etendus. Ibn STna est surtout le philosophe de
langue arabe, et sa logique est encore aujourd’hui enseignee dans les ecoles
musulmanes... [Madkour, 1969 2 , pages 9-10]
The alternative approach assumes that logical differences map precisely onto the dif¬
ferences among philosophical schools; to write a logical history of the realms of Islam,
one need only write their more general philosophical history. Thus we find Rescher
in [Rescher, 1964] tracing the filiations of the philosophical schools, and placing logicians
and their writings onto the genealogy produced—the logicians inherit as proponents or
opponents of their logic the same philosophers who promote or oppose their metaphysics.
As I hope will become clear in the course of this chapter, Rescher’s approach accounts
far better than Madkour’s for the logical texts produced by the various authors; in fact,
I have adopted Rescher’s model as a heuristic device to work with in writing this chap¬
ter. Three changes seem to me to be in order. Firstly, the notion of ‘school' as a way
of collecting groups of logicians is fine, so long as it is recognized that these ‘schools’,
particularly later on, may have had no fixed point of convention, no set curriculum, and no
doctrinal unanimity. They tend to be united only by pedagogical lineage, itself often ten¬
uous. The Baghdad school is closest to being a school in our usual sense of the word, but
528
Tony Street
it is unclear that its teaching practice was common later on and further east. Secondly, the
periodization of Rescher’s study (800 to 900,900 to 1000, and so on) does not even have
the doubtful merit of following the temporal boundaries of ruling dynasties. Abbasid im¬
perial policy began the earliest sustained efforts to translate logical works [Gutas, 1998],
but by the end of the second Abbasid century (950), it was debate about that translated
logical problematic conducted among logicians writing in Baghdad and elsewhere that
determined the further fortunes of logic in the Islamic world. I have tried to structure this
chapter around moments of particularly intense logical activity in the realms of Islam.
Thirdly, and mainly due to his conviction that divisions among logicians follow divi¬
sions among philosophical schools more broadly conceived, Rescher tended to write his
history from biobibliographical accounts of Islamic philosophical history. I have supple¬
mented his account with different sources. I try mainly to use the references logicians
made in their writings to other logicians. Post-Aristotelian logicians generally speaking
are given to referring to other logicians, and the logicians writing in Arabic are no ex¬
ception. These references serve to modify aspects of Rescher’s account. It has to be
stressed, however, that the references I have gathered, taken together, are not sufficient to
do anything more constructive than modify an existing account. But I believe that there
are sufficiently many such references ultimately to produce a far sounder history than we
have at present.
Sources A few words are in order concerning the sources. Ideally, of course, one would
give oneself up, like a corpse to the body-washers, to the vast body of logical treatises
and their varying formulations of different logical doctrines, only regaining critical con¬
sciousness to note the past and contemporary logicians to whom they refer. As a paltry
beginning to that ideal task, I have decided to examine aspects of the syllogistic. I have
chosen the syllogistic due to the concentration of existing scholarship; I do not think that
any other single area in Arabic logic has been studied through so wide a range of writers,
or so successfully, from both a technical and a philological point of view. It may not be
the most interesting of the achievements of the logicians writing in Arabic, but it is central
to their systems, and each one of them treats it.
When I talk about the concentration of scholarship on the syllogistic, I mean especially
the editions we have of the earliest Arabic version of the logic [Danispazuh, 1978], and
the achieved translation of the Prior Analytics [BadawT, 1948/52]; the careful study of
the Prior-Analytics complex, its technical terms, and the use made of the Prior Analyt¬
ics by Alfarabi [Lameer, 1994]; the translations of texts by Avicenna which exposit the
central features of his theory of the syllogistic [Goichon, 1951; Inati, 1981]; the anal¬
ysis of Averroes’ changing treatments of the modal syllogistic [Elamrani-Jamal, 1995],
and the editions and studies of his major extant texts on the subject [Averroes, 1983a;
Averroes, 1983b]; and the description of, and semantics for, the syllogistic system com¬
mon to most writers after the end of the thirteenth century due to Rescher and vander
Nat [Rescher and vander Nat, 1974].
In testing and modifying Rescher’s account, I have looked especially at the logicians
who are writing at historically significant moments, moments when new directions are
claimed to be either beginning or ending. For this reason, I have looked especially at Abu-
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1-Barakat al-BagdadT (d. 1165),FahraddTn ar-RazT (d. 1210), NasTraddTn at-TusT (d. 1272),
and Qutbaddln ar-RazT at-TahtanT (d. 1365). Throughout, I have written with an eye to
explaining how the widely used treatise by Nagmaddln al-KatibT (d. 1276 or 1294), Logic
for Sarnsaddm [KatibT, 1854], came to acquire the form and content it has. This has
helped to shape the narrative of this chapter and the way I read all the logicians consid¬
ered. Inevitably, people will object to a number of the sources I have not used (especially
works by Alfarabi and Averroes), the writers I have neglected, and the range of logical
disciplines in the tradition which I have ignored. For those who want to know in advance
what is not treated in this chapter, section 6 provides a summary of its main points, and in¬
dicates some of the lines of research not broached here. I hope the bibliographical notes in
the third appendix help direct people disappointed in this way to further relevant material
or, at least, to lists of such material.
2 THE TRANSLATION OF THE ORGANON
For more than two centuries throughout the period of the Graeco-Arabic translation move¬
ment the Organon was translated and revised numerous times by succeeding generations
of scholars in accordance with their philosophical and philological needs. By the time a
settled version had been achieved, a number of other commentaries by Greek writers of
late antiquity had also been translated to help make sense of it. In this section of the chap¬
ter, the history of the translators and their cumulative efforts in rendering the Organon
and related works into Arabic will be presented in four stages: the Syriac translations,
the earliest Arabic translations, the translations produced by the translation circles headed
by al-Kindl and by Hunayn, and the later revisions. It must be stressed, however, that the
process was not in fact a linear progression to a final and complete version of the Organon,
but was much more fluid, being carried out by a number of translators with varying lev¬
els of technical skill and differing philosophical priorities. Some of the early translations
found a permanent place among the writings Arabic logicians read as a matter of course
in centuries to come, while other translations were subject to revision after revision.
The Syriac translations A great many of the early translators of logical works were
Christians belonging to one or other of the Syrian Churches. This is because these
churches had for many centuries taught some logic from the Organon. That said, they
taught even less than had been taught in the Alexandrian curricula of the sixth century,
and limited themselves to the Categories, On Interpretation, and the assertoric syllogistic
in the Prior Analytics (which is to say, to the end of the seventh book of the first part).
The reason Alfarabi gave for their stopping at that point was this:
The Christians stopped instruction in Rome, but it carried on in Alexandria,
until the Christian king looked into it. The bishops gathered and took counsel
on what part of philosophical instruction should be left, and what should be
stopped; they came to the opinion that the logical texts up to the end of the
assertoric figures should be taught, but not what comes after that. They came
to this opinion because they thought that the later parts were injurious to
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Christianity, while the earlier parts that they permitted contained things that
would help towards the promotion of their religion. So this amount remained
in public instruction; the rest was looked into privately, until Islam came, a
long time after... ([Ibn-AbT-Usaybi‘a, 1882, page 135.8-14]; cf. [Lameer,
1994, page xviii], and especially [Gutas, 1999, pages 163 & 164])
It has been suggested that it was the same reason behind the narrow compass of the old
logic in the West [Pines, 1996]. Whatever the historical merits of this argument, it must be
borne in mind that the Syriac tradition always had the early parts of the Prior Analytics,
and that provided textual authority against the kind of speculation about conversion that
a scholar like Abelard was free to pursue. The old logic of the Syriac churches had
potentialities different from the old logic of the West. It is even more important to note
that the story of the Christian interdiction on the study of modal logic and beyond bears
the marks of polemical tensions that pre-date Alfarabi. Once these are cleared aside, we
can discover that Syriac logic was a constriction on the Alexandrian logical curriculum;
when logical studies were widened in Baghdad at the turn of the tenth century to cover
the whole of the Organon, it was a major structural change indeed ([Gutas, 1999, page
186]).
By the sixth century, a decline in the knowledge of Greek among Syriac Christians
meant that any part of the Organon and related logical writings that were to be studied
had to be translated into Syriac. One of the greatest translators in the resulting translation
movement was Sergius of Res‘ayna(‘the Boethius of the Syriac tradition’; d. 536); he was
followed by important commentators such as Severus Sebokht (d. c. 666) and Athanasius
of Balad (d. 686) [Brock, 1993]. These men were among the first to attempt the difficult
task of translating the Organon into a Semitic language. Subsequent translations of the
Organon into Arabic almost always went by way of a prior Syriac translation.
The earliest Arabic translations The early Caliphate and the Umayyads neither dis¬
couraged nor advanced logical studies in their newly conquered territories. But in 750,
the Abbasid dynasty came to power, and by 756 had founded the empire’s new capital,
Baghdad. For various reasons, translation activity served to further Abbasid imperial pro¬
paganda and was therefore encouraged. The translation movement began by drawing on
the living Syriac pedagogical tradition in philosophy and, to a lesser extent, the Sassanian
tradition, but it soon came overwhelmingly to surpass both these traditions in terms of
range and quality of the translations it produced and the energies it devoted to the teach¬
ing and study of those works. In fact, the translations created and met cultural needs
in such a way that they came to be sustained alongside and ultimately without official
Abbasid support [Gutas, 1998].
For all the importance of the Syriac Christians in the movement, it may well be that it
was a Zoroastrian convert to Islam who produced the first work on logic written in Arabic.
Ibn-al-Muqaffa‘, who was executed in 757 for political reasons, translated an epitome of
the Categories, On Interpretation and the first part of the Prior Analytics, and prefaced
the whole work with a short introduction on the value of philosophy and a brief treat¬
ment of the predicables [Danispazuh, 1978, pages 1-93], Ibn-al-Muqaffa‘ was not the
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531
only scholar to translate this short textbook ([Danispazuh, 1978, page 93]; cf. [Lameer,
1994, pages 11-12]), but we have no clear idea about who wrote the original. What¬
ever its origin, however, in covering the categories, ending with the assertoric syllogistic,
but stretching to include some coverage of the Posterior Analytics, the work belongs to
the pre-Syriac Alexandrian tradition ([Gutas, 1999, pages 184 & 185]). Soon after this
textbook became available, a translation of the Topics was commissioned by the Caliph
al-Mahdl (d. 785), and the commission was carried out by the Nestorian Patriarch Timo¬
thy I and Abu-Nuh, a Christian secretary, working with a Syriac intermediary text [Gutas,
1998, page 61]. The way the Topics was translated became typical. Among other schol¬
ars working at these early translations were Theophilus of Edessa (d. 785), who worked
as the Caliph’s court astrologer, and Theodore abu-Qurra (d. 826), the Melkite Bishop
of Harran. The Christian tradition of which these scholars formed a part continued into
the later period, and included Hunayn ibn-Ishaq (d. 873), Ishaq ibn-Hunayn (d. 910), and
Abu-Bisr Matta (d. 940). t,From the ninth century on, many scholars in the Christian
tradition had come to write in Arabic by preference, and not merely in fulfillment of a
translation contract; such scholars include Yahya ibn-‘AdT (d. 974) and ‘Abdallah ibn-at-
Tayyib (d. 1043) [Brock, 1993, page 9].
The translation circles How the translators were employed is not entirely clear. A few,
like Theophilus of Edessa, were employed directly by the Caliph. Most seem to have
enjoyed the patronage of courtiers and wealthy patrons, or even just worked on projects
by contract. They formed groups, perhaps only loosely affiliated. By around the 840s,
the first famous circle of translators had formed around the celebrated ‘philosopher of the
Arabs', al-Kindl (d. 873) and his student, as-SarahsI (d. 899). Although Kindi and Sarahs!
were Muslims, most of their colleagues were Christians. The circle seems to have been
interested in texts we now associate with neoplatonism [Zimmermann, 1986], and one
member of the group, Ibn-Na‘ima (fl. c. 830). translated the Sophistical Fallacies as well
as the Plotinian Theology of Aristotle. This did not exclude interest in Aristotle, of course,
and Kindi himself wrote an outline of the Organon [Kindi, 1950; Rescher, 1963b].
Due to recent important work by Endress [END, ], we have a rich understanding of
the activities and goals of the Kindl-circle. It was probably fairly loosely constituted, and
the fact that Kindi revised a translation by Ibn-ai-Bihrlz (d. c. 860), the Bishop of Mo¬
sul (who also produced the earliest surviving compendium of logical terms [Danispazuh,
1978, pages 97-126]), may not indicate that Ibn-al-Bihrlz was actually part of the circle.
In fact, Ibn-al-Bihrlz enjoyed the patronage of the Buhtlsu‘ family, one of the wealthy
families supporting translation work, and we know that the Buhtlsu 1 family particularly
supported the circle working around Hunayn ibn-Ishaq (d. 873). Hunayn was a Nesto¬
rian Christian, and the most famous of the translators. The dates at which he and Kindi
were working must have overlapped. He, his son Ishaq ibn-Hunayn (d. 910), and his
pupils translated nearly the whole of the Organon. Hunayn and his pupils drew on ear¬
lier Syriac translations when available, and on the Greek commentaries of late antiquity.
Hunayn’s primary interests were medical, and he held Galen in high regard; in conse¬
quence, he translated many of Galen’s logical works along with the medical works, in¬
cluding the Institutio Logica , a treatise on the number of syllogisms, fragments dealing
532
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with On Interpretation, and fragments of On Demonstration [Bergstrasser, 1925, pages
47^)8]. Galen may well have dominated logical studies in Baghdad for one or two gen¬
erations after Hunayn, but a reaction ultimately set in, and neither Alfarabi and Avicenna
acknowledged any debt to Galen’s logical works ([Zimmermann, 1981, page lxxxi]; this
does not mean that they did not share some of his ideas; see [Shehaby, 1973b, pages 5 &
6] and [Lameer, 1994, pages 10, 47]).
The period of revision The activities of the group of scholars who had worked around
Hunayn and his son carried on after Ishaq’s death, and merged seamlessly with the ac¬
tivities of a new group of scholars in Baghdad who worked with the texts more critically
and philosophically. Claiming to represent the true pedagogical lineage of the Alexan¬
drian school, a certain Abu-Yahya al-MarwazT taught the scholars who taught the great
luminaries in the Baghdad school of the early tenth century: Abu-Bisr Matta ibn-Yunus
(d. 940), and Alfarabi (d. 950).
Alfarabi claimed himself to be an Aristotelian and, in making this claim, he meant to
be taken as a true Peripatetic, doing something rather more rigorous than his predecessors.
This is not to say that Alfarabi did not take a great deal from the Syriac tradition. Among
other things, the sources for the logic chapter of his Enumeration of the sciences are
mediated through that tradition [Gutas, 1983, page 255 ff.]. But Alfarabi came to respone
to two urges in his work as a logician. Firstly, as a Muslim, he felt a desire to explain the
whole enterprise of logic in terms that exponents of the other Islamic disciplines would
understand. Secondly, he became increasingly aware that the introductory Syriac treatises
obstructed understanding of the original Aristotelian texts, and he wanted to rectify this.
What matters for present purposes is the second of these pressures. Alfarabi began a
critical examination of the Aristotelian texts, often setting aside the prevailing interpre¬
tation. This must have galled some of his colleagues, and they seem to have cited him
less than one would expect given the quality of his work ([Zimmermann, 1981, page cxi
& note 1]; but see also [Marmura, 1983, page 763b]). Later historians of philosophy
referred to Alfarabi as a leading philosopher, a ‘head of school’, distinguished by his crit¬
ical attitude towards and interaction with the translated texts. His colleague Abu-Bisr,
and one of his students, Yahya ibn-‘AdT (d. 974), were similarly distinguished, as were
Yahya ibn-‘Adi’s students, Abu-Sulayman as-Sigistanl (d. c. 990), Ibn-Zur‘a (d. 1008),
and Ibn-Suwar (d. 1017).
With Ibn-Suwar, we may say that the translation and refinement of the Organon was
complete. It is his copy of the Organon, with extensive marginal and interlinear notes,
which was copied as the manuscript now in the Bibliotheque Nationale (codex ar. 2346),
and which serves as the basis of our contemporary edition ([BadawT, 1948/52]; cf. now
[Jabre, 1999], which further uses MS Istanbul Ahmet III 3362). Ibn-Suwar is a product
of the Baghdad approach to the Organon, and his version of it bears elegant testimony to
the intimacy of the connexion between translation and interpretation, philology and phi¬
losophy. For a long time, the interlinear notes and marginalia of codex ar. 2346 have been
recognised as showing the growing philological acuity of the Baghdad school [Walzer,
1962]; what they also show is a sophisticated philosophical reaction to the text as it was
being received into Arabic ([Hugonnard-Roche, 1993]; but see also [Lameer, 1996]).
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533
We are fortunate to have from this time the catalogue of a Baghdad bookseller, Ibn-an-
NadTm (d. 995), called simply The Index, completed in 987 ([Ibn-NadTm, 1871/72]; the
logic chapters are translated in [Peters, 1968]). Ibn-an-Nadlm knew Ibn-Su war personally,
as well as other members of the school of which he was a part. The Index provides most
of the information we have about the translators of the various parts of the Organon and
of the various Greek commentaries on it that were also translated. Aside from the works
of Galen, these Greek commentaries included works by Theophrastus, Porphyry, John
Philoponus, Stephanus the Alexandrian, Ammonius, Themistius, Simplicius, Iamblichus,
a mysterious AlTnus [Elamrani-Jamal, 1989-] and, perhaps most importantly, works by
Alexander of Aphrodisias. <,From The Index one also gets a sense of the nature of the
shared enterprise that produced Ibn-Suwar’s version of the Organon: draft translations,
commentary, discussion, revised translations. Through this process, the translators also
established a technical vocabulary with which to render the Organon [Afnan, 1964], it¬
self one of the great achievements of the translators. This technical vocabulary, and the
translations in their various stages of refinement, enabled and were in turn enriched by
vigorous philosophical debate.
A note on the Arabic Prior Analytics Lastly, since this chapter focuses on the syllogis¬
tic, especially as presented in the first twenty-two books of the Prior Analytics, it is worth
making a few specific comments about the translation of the Prior Analytics. The trans¬
lator was the mysterious TadarT, plausibly identified by Lameer as a certain Theodore,
a Syriac Christian working with Hunayn’s circle [Lameer, 1994, pages 3 & 4], though
quite possibly working long before that time. TadarT was making his translation against
a background in which the syllogistic was known at least superficially due to works like
the text by Ibn-al-Muqaffa‘. His translation was corrected by Hunayn, and, somewhat
strangely for the translations of the Organon, used as a basis for the Syriac translation by
Hunayn and Ishaq [Ibn-NadTm, 1871/72], The edition which we have [BadawT, 1948/52,
vol. I, pages 103-306] is based on the Paris manuscript, and the parts directly relevant to
this chapter are Prior Analytics 24 a 10-40 6 16 & 50 6 [BadawT, 1948/52, pages 103-176
& 217-218]. ([Jabre, 1999] came to my attention too late to be used for this chapter.)
Some reflections on the notes that would have been available to Alfarabi, and his attitude
to them, are given in Zimmermann [Zimmermann, 1981, pages lxxiv-lxxv]; the Prior-
Analytics complex has been reconstructed as far as is possible by Lameer [Lameer, 1994,
chapter one]. One general comment can be made: the Prior Analytics came with a mass
of interpretative material, not of all of which was mutually compatible.
3 ALFARABI AND AVICENNA
Alfarabi (d. 950) and Avicenna (d. 1037) are the two most important writers to consider
in constructing a history of medieval Arabic logic. They constitute, with Aristotle, the
three main reference points for later writers as they react against or conform with the
philosophical options before them. Both Avicenna and Alfarabi came to the Baghdad
translation of the Aristotelian corpus by way of the translated commentatorial material.
534
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As they worked, both sought to set themselves apart from the bulk of the Syriac Chris¬
tians who, numerically at least, had dominated philosophising in the Islamic world up to
that point. In this, Avicenna was explicitly following Alfarabi as his venerated forebear.
This is true even though Avicenna differed from Alfarabi at many points, especially in
the logic. With Avicenna’s awarding Alfarabi a pre-eminent place in the history of Aris¬
totelian philosophy, and with his reformulation of the Aristotelian system, we may say
that a truly naturalized tradition of logic in the realms of Islam begins.
We know hardly anything about Alfarabi. He was born somewhere in the East, perhaps
in Transoxiana, perhaps around 870. He moved at some time to Baghdad, and it was
probably there that he studied logic up to the end of the Posterior Analytics with Yuhanna
ibn-Haylan. This took place some time within the reign of al-Muqtadir, which is to say
between 908 and 932. Near the end of 942 he left Baghdad for Syria, and worked in
Damascus and Aleppo. He visited Egypt towards the end of his life, then returned to
Syria, and died in Damascus in 950 or 951 [Gutas, 1982—b]. Most of his surviving works
are on logic, or deal with logic as a central theme.
By contrast, we know quite a lot about Avicenna, and we can be confident that most
of what we know is accurate. Born in Bukhara some time before 980 [Gutas, 1987-88],
Avicenna spent the first twenty years of his life on philosophical studies, most of which
he undertook without a teacher. Faced with political upheaval in his homeland, Avicenna
travelled from one principality to another: from Bukhara to Khwarezm, after some years
on to Jurjan, then to Rayy, to Hamadhan, and finally to Isfahan. In each of these places, he
supported himself by his skills in medicine and administration. He died in 1037, leaving
behind a huge corpus of works, many of which deal with logic.
3.1 Approaches to the Aristotelian tradition
The ways Alfarabi and Avicenna approach the logic are determined by their attitudes to
the broader Aristotelian tradition. Each of them wrote a short text designed to clarify his
attitude to Aristotelianism.
Alfarabi’s approach Alfarabi is part of a movement in Baghdad which began around
900 with his teacher Yuhanna ibn-Haylan and his senior colleague, the Christian Abu-Bisr
Matta. Alfarabi described his philosophical pedigree in a short tract, On the appearance
of philosophy, in which he claimed his teachers and himself to be Aristotelians, alumni
of the Alexandrian school, a school whose move to Baghdad he also traced in the tract.
There are many difficulties in reaching a good understanding of the considerations that
went into the composition of this tract (see now [Gutas, 1999]), but it is possible to state
concisely what being an Aristotelian meant for Alfarabi’s conception of his forebears.
If, as would appear, the pillars of the Baghdadian renaissance were Alexan¬
der and Themistius, neither of whom had been connected with Alexandria,
it makes sense to talk of a continuity of Alexandrian tradition only in al-
Farabi’s scheme of the history of philosophy according to which all Greek
Aristotelians, on the strength of their spiritual connexion with the legendary
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535
school of Aristotle at Alexandria, would qualify as representatives of ‘Alexan¬
drian’ tradition. [Zimmermann, 1981,civ-cv]
For present purposes, the historicity of Alfarabi’s On the appearance of philosophy is be¬
side the point; what matters is that Alfarabi was consciously trying to revive a true, textual
Aristotelianism after a period of rupture [Hasnawi, 1985]. He was doing this, moreover,
without making any mention of Kindi and his circle, or of the sui generis Muhammad ibn-
Zakariya ar-RazT [Gutas, 1999, page 155]—Alfarabi obviously considered these philoso¬
phers part of the problem.
Another motivation behind many of Alfarabi’s formulations was his consciousness of
working in an Islamic community. At the time that Alfarabi was studying and teaching
in Baghdad, the various Islamic disciplines were achieving their classical articulation.
Alfarabi worked towards both making philosophy resemble the Islamic disciplines in its
historical claims, and making its utility for and complementarity with those disciplines
obvious [Gutas, 1982-a, page 219].
Avicenna’s approach Avicenna was just as ardently Aristotelian as Alfarabi, but his
Aristotelianism was constituted and implemented in different ways. It was constituted
differently in that Avicenna’s respect for Aristotle was not alloyed with a correspondingly
high respect for Plato: “if the extent of Plato’s achievements in Philosophy is what came
down to us of him, then his wares were paltry indeed and philosophy in his time had not
matured to the point of reaping” [Gutas, 1988, page 38]. Further, as has been mentioned,
Avicenna had woven Alfarabi into his litany of great past philosophers, and reassigned
lesser positions to some members of the Greek schools; on his work at one point in his
later life, he wrote:
... [I] am occupied with men like Alexander [of Aphrodisias], Themistius,
John Philoponus, and their likes. As for Abu-Nasr al-Farabl, he ought to be
very highly thought of, and not to be weighed in the same scale as the rest:
he is all but the most excellent of our predecessors. ([Gutas, 1988, page 64];
cf. [BadawT, 1948, page 122.2^1])
Avicenna’s respect for Alfarabi was joined to an explicit contempt for the Syriac Chris¬
tian philosophers. One of Avicenna’s students remembered in his memoirs that Avicenna
condemned Ibn-Suwar (whose version of the Organon is so important for modern schol¬
ars; see above page 532) and his colleagues, who, “because their field is so narrow, adhere
more closely than others to the [traditional] transmissions of certain books.”
Upon my life, these people relax and are satisfied with whatever they imagine
to be the case which is easily treated, dismissing logic absolutely. With regard
to the matters of syllogisms, their dismissal is complete and they pay no
attention whatever to them—and not only today, but they have been doing
this for quite some time. As for the forms of syllogisms, specifically these
people have disregarded them. Whenever they treated them, they strayed
from the right path because they never acquired the habit of dealing with
them and they never suffered the pains of analyzing the details of problems
536
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so that they may gain a syllogistic habit; their sole reliance, instead, is upon
ideas not subject to rules. [Gutas, 1988, pages 68-69]
It is in the implementation of his Aristotelianism, however, that Avicenna differed more
significantly from Alfarabi. Whereas Alfarabi constructed the story of his philosophical
education to tie himself to a school and its teaching tradition, Avicenna constructed the
story of his education to sever himself from any teaching tradition at all. He designed
his autobiography to present himself as an autodidact successful by virtue of intuition.
Gutas has summed up the major effect that the doctrine of intuition which undergirds the
autobiography has for Avicenna’s reading of Aristotle.
The perspective of the Autobiography, therefore, is that of a philosopher be¬
longing to no school tradition, who established truth on his own by means
of his Intuition, equalling Aristotle in this regard, if not surpassing him, and
whose independent Verification of the truth, which reproduces for the most
part the philosophical sciences as classified originally by Aristotle, puts him
in a position both to teach this more accurate version of the truth, and to judge
the philosophical attainments of others. [Gutas, 1988, pages 197-198]
The content of the doctrine of intuition need not concern us here—what matters is that
its effect was to position Avicenna relative to the Aristotelian corpus differently from
Alfarabi. When Avicenna collided with a crux in the text, he did not have to resort to
exegetical strategies to find his way out. In fact, throughout The Cure it is clear that he
believed he had worked out the unified vision that motivates Aristotle’s presentation, and
this allowed him to elide, transform and augment the system of the Prior Analytics.
3.2 Alfarabi and the logical treatise
Alfarabi’s attitude to Aristotle seems to have become clearer over time, and in conse¬
quence his position changes from one work to another. And Alfarabi had any number of
opportunities to change his position: he wrote many works, nearly half of which seem
to have been addressed principally to logic. In some places we find undigested stretches
of logical doctrine which do not fit well with the rest of what he is doing [Zimmermann,
1972]. Further, we find that from one logical treatise to the next, some terms are be¬
ing used more precisely, others are being discarded, and doctrines are being clarified as
their relevance to each part of his project is established [Lameer, 1994, e.g pages 202,
259-289]. Alfarabi writes in the tradition of the Alexandrian summary, as a Muslim in
an Islamic society, and as an Aristotelian concerned to recover the true sense of Aris¬
totelian texts. Only the first two activities will be considered in this section—Alfarabi the
Aristotelian commentator is considered in sections 3.4 and 3.5 below.
The fact that Alfarabi’s final views on logic only came to be delineated over time makes
it difficult to describe his logic generally. Further, we have lost a number of his works,
especially the long commentaries. Most significantly for the line of investigation I am
following in this chapter, we have lost the first section of the Long Commentary on the
Prior Analytics , although we can reconstruct enough of this work for present purposes
Arabic Logic
537
from references in Avicenna and Averroes. We do have the long commentary to On
Interpretation , and shorter commentaries on the other books of the Organon; and we
have many works in which logic is a major or the major subject under discussion (for
a short overview, see [Gutas, 1993, pages 47-50]; cf. [Lameer, 1996, page 97]). But
these works were addressed to various audiences, and it is not always easy to say what
the relative importance is of various doctrines, nor, because so many of the longer works
are as yet unavailable, whether Alfarabi at the stage of his most mature reflections would
have wanted to affirm any given doctrine. In short, we are not now, and probably never
will be, able to describe Alfarabi’s logic with confidence. Still, we can name a number of
doctrines later logicians adopted from him.
Logic and language Alfarabi presented his definition of logic by contrasting it with
grammar. To hold that this is the best way to go about such a definition is an important
philosophical claim, which Avicenna was later to reject; but it is also a political claim,
finding a position for Greek logic within Islamic society. Grammar is not uncommonly
contrasted with logic—it is implicitly contrasted with logic and rhetoric throughout the
medieval Western tradition of the trivium—but events in Alfarabi’s Baghdad had made
such a contrast especially urgent. Alfarabi’s senior colleague, Abu-Bisr Matta, had been
ignominiously routed by a grammarian, Abu-Sa’Td as-STraff (d. 978), who doubted the
scholarly viability of an independent subject like logic given that people had grammar (see
below page 554). Subsequent treatments of logic tended to inherit from discussions like
this an apologetic edge, trying to find a task for logic separate from but complementary
with grammar. Alfarabi wrote in his Introductory epistle on logic:
Our purpose is the investigation of the art of logic, the art which includes the
things which lead the rational faculty towards right thinking, wherever there
is the possibility of error, and which indicates all the safeguards against error,
wherever a conclusion is to be drawn by the intellect. Its status in relation to
the intellect is the status of the art of grammar in relation to language, and
just as the science of grammar rectifies the language among the people for
whose language the grammar has been made, so the science of logic rectifies
the intellect, so that it intellects only what is right where there is a possibility
of error. Thus the relation of the science of grammar to the language and the
expressions is as the relation of the science of logic to the intellect and the
intelligibles, and just as grammar is the touchstone of language where there is
the possibility of an error of language in regard to the method of expression,
so the science of logic is the touchstone of the intellect where there is the
possibility of an error in regard to the intelligibles. [Dunlop, 1956, page 230
(Arabic, page 225)]
So grammar deals with the manipulation of expressions in a particular languages, whereas
logic deals with the manipulation of meanings common to all peoples. Alfarabi’s general
point was accepted among his colleagues in Baghdad, though the extent to which the
intelligibles can be compared to separate utterances was disputed, especially by Avicenna.
538
Tony Street
The context theory In the Introductory epistle we also find Alfarabi presenting his logic
according to what is known as the context theory, which he inherited from the Alexandrian
tradition. Thus he took the Categories, On Interpretation, and the Prior Analytics to have
general application across all stretches of discourse, and each of the following five books
of the Organon to have only specific utility for a particular mode of discourse. (Although
he did accept the Categories as a logical work, Alfarabi recognised the force of arguments
that sought to classify it as metaphysical.)
According to the context theory, syllogisms with premises of differing epistemic grades
constitute distinct stretches of discourse, and may belong to demonstrative philosophy,
dialectic, sophistry, rhetoric or poetry (cf. [Gutas, 1983, pages 256-257 and diagrams IV
& V]). That is to say, syllogistic contributes to the analysis of every stretch of discourse.
Syllogism is employed either in discoursing with another or in a man’s bring¬
ing out something in his own mind... Philosophical discourse is called demon¬
strative. It seeks to teach and make clear the truth in the things which are such
that they afford certain knowledge. Dialectical discourse seeks to overcome
the interlocutor in the things which are known and notorious. Sophistic dis¬
course seeks to overcome the interlocutor by a supposed victory in the things
which are thought ostensibly to be known, without being so. The aim is to
draw the interlocutors and hearers into error, likewise falsification and trick¬
ery, and that the speaker should produce the opinion of himself that he is
one who possesses wisdom and knowledge, without being so... Rhetorical
discourse seeks to satisfy the hearer by what will partially content his soul,
without reaching certainty. Poetical discourse seeks to represent the object
and suggest it in speech, as the art of sculpture represents different kinds of
animals and other objects by bodily labours. The relation of the art of poetry
to the other syllogistic arts is as the relation of sculpture to the other practi¬
cal arts, and as the relation of chess-playing to the skilful conduct of armies.
[Dunlop, 1956, page 231 (Arabic, page 226)]
The context theory depends on a division of discourse according to the ‘matter’ which
makes it up—divisions of this matter came to be disputed, and even Alfarabi seems to
change his mind from one text to the next. However the material aspects of discourse
were divided, each of the resulting divisions was related to one or other of the faculties
of the soul, and can be explained fully only in tandem with a treatment of psychological
and epistemological doctrines ([Black, 1990, chapters 4 & 6]; cf. [Lameer, 1993]). The
extent to which the syllogistic contributed to the analysis of various stretches of discourse
was also disputed, and Avicenna doubted that rhetoric or poetics could really be treated
formally as syllogistic. The context theory and its permutations continued to be a factor
in logic manuals at least until the seventeenth century, though not always dictating the
same structure in each treatise. It also had consequences for how logic related to the
Islamic disciplines, such as those devoted to the analysis of dispute, and to rhetoric; as the
Islamic disciplines came to exert a stronger claim to these fields, their treatment within
logical treatises became sketchier and sketchier (see Ibn-Haldun’s brief comments on this
phenomenon, page 580 below).
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539
Syllogistic analysis of arguments In Alfarabi's works we find another feature which
recurs in a great many Arabic logical texts: the attempt to show that all valid argument-
forms relate in some way to the syllogism, an attempt first made by Aristotle. Like the
claim that logic and grammar had different but complementary interests in language, the
claim that the structure of the syllogism was important in understanding the structure
of other arguments was directed at Muslim jurists and theologians. In this effort, the
earliest relevant text Alfarabi wrote was The short treatise on reasoning in the way of the
theologians
... in which he interpreted the arguments of the theologians and the analogies
(qiyasat) of the jurists as logical syllogisms in accordance with the doctrines
of the ancients. ([Alfarabi, 1986b, page 68.11-12]; cf. [Sabra, 1965, page
242a])
In this text, we find analyses of the paradigm, of the argument used by Muslim theolo¬
gians called ‘reasoning from the seen to the unseen’ ( al-istidlal bis-sahid 'ala l-ga’ib),
and of ‘the juristic argument’ ( al-qiyas al-fiqht) itself. Alfarabi takes the second kind of
argument to reduce to the first, and offers an elaborate analysis of the third as involving
a range of rhetorical argument techniques [Lameer, 1994, respectively, chapters 6,1 &
8]. This began a trend which did in fact issue in the acceptance of logic as useful by
an important Muslim jurist, Abu-Hamid al-GazalT (see below page 554 ff.). It also led
subsequent writers of logic manuals to consecrate at least a part of their manuals to the
reduction of argument-forms to the syllogism, a reflex carried over from this early time
when Muslim scholars contested the place of logic in Islamic society.
3.3 Avicenna and the logical treatise
At the latest, Avicenna came by his middle age to a settled view of the proper conception
and formulation of logic. Like Alfarabi, a large proportion of his work was given over to
logic (for a brief overview of his works and their genres, see [Gutas, 1993, pages 50-53]).
Though we lack any of his early commentaries directly on the texts of Aristotle, we have
all of The Cure , Avicenna’s great philosophical opus. The first book of The Cure treats the
subject-matter of Porphyry's Introduction, and each one of the next eight books covers the
subject-matter of each of the parts of the Alexandrian arrangement of the Organon. We
also have a number of shorter expositions, three of which I refer to in this chapter. Two
of these shorter expositions. The Book of Salvation and Pointers and Reminders, present
the system with all the sophistication we find in The Cure , while the third, in Philosophy
for ‘Ala’uddawla , presents a greatly simplified system.
Avicenna’s books became important as paradigms for subsequent writers. So, for ex¬
ample, we find Abu-l-Barakat al-BagdadT (d. 1165) consciously modeling his major philo¬
sophical work on The Cure. Most important of all Avicenna’s works, however, is Pointers.
Because of its difficult and allusive style, it became the subject of many commentaries—
these evolved in time into free-standing treatises which none the less preserved the order
and emphases of Pointers. Many of the changes in the treatment of logic which Ibn-
Haldun notes (see below page 580) are apparent already in Pointers.
540
Tony Street
Rejection of Farabian doctrine In defining logic, Avicenna differed from Alfarabi.
Avicenna agreed that logic was a normative instrument to protect man from going astray
in thinking ([Avicenna, 1971 2 , pages 117-127]; cf. [Gutas, 1988, page 281]). But he did
not characterise logic in the way Alfarabi did.
There is no merit in what some say, that the subject-matter of logic is spec¬
ulation concerning the expressions insofar as they signify meanings... And
since the subject-matter of logic is not in fact distinguished by these things,
and there is no way in which they are its subject-matter, [such people] are
only babbling and showing themselves to be stupid. ([Black, 1991, page 54];
cf. [Avicenna, 1952, pages 23.5-6,24.3^1])
One reason for this is that in Avicenna’s psychology, language as a set of discrete expres¬
sions is not essential for the intellect in its operations; it is only accidentally the path that
humans have to follow.
... [I]f it were possible for logic to be learned through pure cogitation, so
that meanings alone would be observed in it, then this would suffice. And it
if were possible for the disputant to disclose what is in his soul through some
other device, then he would dispense entirely with its expression. ([Black,
1991, pages 54-55]; cf. [Avicenna, 1952, page 22.14-17])
In consequence, intelligibles are not able to be likened to expressions in a language, which
must by their essence be uttered and grammatically ordered through time.
Modifications to Alfarabi’s doctrines Avicenna differed from Alfarabi in holding that
logic does not deal with expressions in so far as they signify meanings. Rather, according
to Avicenna, logic deals with meanings which classify meanings-—logic does not deal
with a proposition’s subject in terms of the meaning it signifies, but as a subject-term.
This is the famous doctrine that the subject-matter of logic is the second intentions (and
here I quote the only doctrine of a medieval Arabic logician that is given in Kneale and
Kneale [Kneale and Kneale, 1962, page 230]):
As you have known, the object of logic is the second intentions ( al-ma ‘am al¬
ma ‘quia al-thaniya )—those that depend upon (tastanid ila) the first intentions—
insofar as they may be of use in arriving at the unknown from the known,
and not insofar as they are thoughts { ma'qfila ) having an intellectual exis¬
tence that is not attached to matter at all or attached to non-corporeal matter.
([Sabra, 1980, page 753]; translation modified slightly)
In this way, Avicenna was able to define logic not only as a normative instrument, as noted
above, but also as an independent science with its own subject-matter, namely, the second
intentions. For all the strong language used in clearing Farabian teaching away to make
space for this doctrine, it would appear that Avicenna is in fact developing ideas found in
Alfarabi [Sabra, 1980, pages 755-756],
It is fairly easy to compare Avicenna with Alfarabi on some other points. In terms
of traditional allegiances, Avicenna was much more forthright in dismissing the logical
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541
writings of the Syriac Christians than Alfarabi had been (see page 535 above). Avi¬
cenna echoed Alfarabi in questioning the propriety of placing the Categories within the
Organon, and decided that it should only be treated within the other logical texts due to
immemorial custom. But it is no help in understanding the syllogism:
The student of logic, after learning what we have told him about regarding
the simple terms, and learning the noun and the verb, can go on to learn
propositions and their parts, and syllogisms, and definitions and their kinds,
and the matters of syllogisms and the demonstrative and non-demonstrative
terms and their genera and species, even if it does not occur to him that there
are ten categories. ([Avicenna, 1959, page 5.1-4]; cf. [Gutas, 1988, page
265])
It is worth noting that in this decision, and in his excision of the assertoric syllogistic (see
below page 548), Avicenna was cutting out of his logic the two things to which the Syriac
Christians devoted most of their efforts.
On the other hand, like the Syriac Christians, and like Alfarabi, Avicenna at the begin¬
ning of his career accepted the context theory, though he dispensed with it later on ([Gutas,
1988, page 18, note 6]). In Pointers, however, he placed the major consideration of the
material aspects of discourse, and its consequences for dividing kinds of discourse, at the
end of his treatment of propositions [Avicenna, 1971 2 , pages 341-364]. This became the
standard place and way to treat the context theory in short treatises thereafter.
Avicenna’s elimination of the categories from his logic texts, and the method by which
he dealt with the context theory, were both influential. But perhaps most important for the
structuring of logical treatises after him was a distinction he found in Alfarabi and used
in his own writings: the distinction between tasawwur and tasdTq. This is a distinction
dividing knowledge into ‘conceptions’ and ‘judgements to which one assents’. Alfarabi
mentioned the distinction in his treatment of demonstration, writing that knowledge “is
of two kinds, conception and assent” [Alfarabi, 1986a, page 19.5], and later implicitly
assigning the logical operations of definition and syllogism to the attainment of, respec¬
tively, conception and assent [Alfarabi, 1986a, page 45.1]. In all of Avicenna’s writings,
by contrast, the distinction is made at the very outset. Among conceptions are for example
‘house’ and ‘man’, and so forth. Among judgements to which one assents are included
for example the judgement that a house is where people dwell, and that man is a ratio¬
nal animal. Indeed, all knowledge is either conception or assent. All investigations are
directed
either to a conception to be acquired, or to an assent to be acquired. It is
customary to call the thing which leads to the desired conception an explana¬
tory phrase, which includes definition and description and the like; and the
thing which leads to the desired assent a proof, which includes syllogism and
induction. [Avicenna, 1971 2 , pages 136-137]
Logic then is concerned to prevent one going astray in thinking about conceptions and
assent; that is, it provides a theory of definition, and a theory of proof [Sabra, 1980, page
761].
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Tony Street
Finally, like Alfarabi, Avicenna agreed that the argument-forms used in law and theol¬
ogy were best analysed by reference to the syllogism ([Avicenna, 1971 2 , pages 365-373];
see also [Avicenna, 1971, pages 38—40]). But unlike Alfarabi, apologetics for logic rela¬
tive to the Islamic disciplines are not central to what he wrote.
Points for comparison There are problems in making further comparisons between
Avicenna and Alfarabi. Alfarabi modified his logical doctrines throughout his life, Avi¬
cenna by and large did not; many texts of Alfarabi are missing, whereas we have the whole
of the Avicennan system (even if we haven’t yet worked through it); Alfarabi wrote both
commentaries on Aristotle and apologetics for logic to propitiate the lawyers, Avicenna
wrote neither (at least in later life).
A comparison, then, is difficult. Luckily for the narrow confines of this chapter, how¬
ever, Avicenna directed comments to the Farabian system, presumably as developed in
the lost Long commentary on the Prior Analytics , while he was dealing with important
points in his own syllogistic. Modern scholars have tended to overlook these comments
because Avicenna referred to Alfarabi as ‘the eminent later scholar’ ( al-fadil min al-
muta’ahhirm), and many have thought that by this he meant Alexander of Aphrodisias
(see for example [Maroth, 1989, page 7]). The eminent later scholar is, however, Alfarabi
(see [Danispazuh, 1989, vol. 3, DTbaga 14]; cf. [Averroes, 1983b, page 101.3-5], and
most recently [Street, 2001]). Avicenna referred to Alfarabi while developing his modal
logic, and at one point in developing his hypothetical syllogistic. Because Avicenna dic¬
tated by his changes what were to be the fundamental questions for later logicians, these
are the major points of discussion by Averroes, and so we find the later tradition effec¬
tively evaluating the earlier traditions with reference to this material.
Four of Avicenna’s references to Alfarabi are particularly helpful for comparing the
systems the two men built. The first point of comparison is made somewhat complex
by the fact that the two men meant different things by ‘absolute proposition’ ( qadiyya
mutlaqa) —I return to this below (see page 547)—and Alfarabi argued that an abso¬
lute e-proposition converts as an absolute e-proposition, whereas Avicenna argued that
e-conversion fails for the absolute. This is symptomatic of more fundamental and far-
reaching differences in how the two men went about laying the foundations for their modal
systems. The second reference is to the fact that Alfarabi accepted Barbara LXL (as did
Avicenna). This raises a problem of consistency for Alfarabi relative to the stratagem he
adopted to save the conversion of the two-sided possible proposition—this is the third
important reference to Alfarabi. Lastly, Avicenna rejected an ascription to Alfarabi of a
long text on the hypothetical syllogistic; this reference allows us to put to rest claims of a
missing long treatment by Alfarabi of the hypothetical syllogistic.
Broadly speaking, then, this provides material for a comparison of how the two men
take the modal syllogistic, and how they take the hypothetical syllogistic. I deal with the
hypothetical syllogistic first.
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543
3.4 The hypothetical syllogistic
I stress at the outset that I do not intend to analyse the hypothetical syllogistic. Alfarabi
probably never treated it in enough detail to ground such an analysis, and the differences
between Avicenna and Averroes touch on deep issues which I do not properly understand.
I will only be using debate about the hypothetical syllogistic as one index for the traditions
to which Avicenna and Alfarabi belonged, and for the way later logicians worked within
their respective traditions.
I use ‘hypothetical syllogistic’ loosely here, as a term sufficiently broad to cover the
two quite different approaches of Avicenna and Alfarabi. In translating Alfarabi’s qiyas
sartl as ‘hypothetical syllogistic’, I am prescinding from the debate about better possible
translations (raised because it does not extend to cover as many inferences in Alfarabi’s
usage as ‘hypothetical syllogistic’ does in Alexander’s usage, but rather seems limited
like Galen’s; see [Lameer, 1994, pages 45^16]). Again, in the case of Avicenna, there
is no technical phrase that corresponds directly with ‘hypothetical syllogistic’—some of
his iqtiraniyyat and all of his istitna’iyyat together would constitute what I mean here by
hypothetical syllogistic (for Avicenna’s technical terms, see below page 546; for aspects
of the usage of istitna, see [Gyekye, 1972]). I use hypothetical syllogistic in the same
way Barnes does:
A hypothetical syllogism is an argument at least one of whose premisses is a
hypothetical proposition. A proposition is hypothetical if it is a compound of
at least two propositions...
Hypothetical syllogistic contrasts with categorical syllogistic, for a syllogism
is categorical if all its component propositions are “simple,” i.e., if none is
compounded of two or more propositions. [Barnes, 1985, page 129]
Alfarabi on the hypothetical syllogistic Alfarabi made the following remark regarding
the hypothetical syllogistic in his Long Commentary on On Interpretation:
He (Aristotle) examines the composition of hypothetical (statements) not at
all in this book, and only slightly in the Prior Analytics. The Stoics, on the
other hand, Chrysippus and others, examined it thoroughly to the point of
excess, made a thorough study of hypothetical syllogisms—as Theophras¬
tus and Eudemus had done after Aristotle’s time—and claimed that Aristotle
wrote books on hypothetical syllogisms. But we have no knowledge of any
separate treatment by him (Aristotle) of hypothetical syllogisms in his books
on logic; this (claim) is found rather in the commentaries of the commenta¬
tors who give an account of them (hypothetical syllogisms) on the authority
of Theophrastus only. [Fortenbaugh and others, 1992, page 239]
Similarly, Alexander of Aphrodisias had said in his own short comments on the hypothet¬
ical syllogistic that “no book of his (Aristotle) on the subject is in circulation. Theophras¬
tus, however, refers to them in his own Analytics —and so do Eudemus and some others
of Aristotle’s associates” ([Barnes, 1985, page 125]; cf. [Shehaby, 1973b, page 24, note
544
Tony Street
11]). In short, Alfarabi belonged to a tradition which was unacquainted with the exis¬
tence of a separate, genuinely Aristotelian treatise on the hypothetical syllogistic, and
which seemed reluctant positively to postulate the existence of such a treatise.
We can also say something concrete about Alfarabi’s own hypothetical syllogistic. Al¬
farabi presented in his treatises (and here I take The short treatise on reasoning in the way
of the theologians as an example) a definition of the syllogism:
A syllogism is a phrase, composed of propositions laid down from which, if
so composed, some other thing follows of necessity by virtue of these very
things themselves, and not accidentally. And whatever comes to be known
through a syllogism is called a ‘conclusion’ or ‘what follows’...
The least from which a syllogism may be composed is two propositions shar¬
ing in a single part; and syllogisms may be composed from hypothetical or
categorical propositions. ([Alfarabi, 1958, page 250.12-apu]; cf. [Lameer,
1994, pages 16-17])
Alfarabi delivered as his hypothetical syllogistic the five Stoic indemonstrable inference
schemata ([Alfarabi, 1958, pages 257.6-260.10]; cf. [Lameer, 1994, page 45]), and did
not take it to contribute to the analysis of the deduction involving a contradiction ([Al¬
farabi, 1958, pages 260.11-261.7]; cf. [Lameer, 1994, pages 50-54]).
It has been speculated that Alfarabi’s lost first part of the Long Commentary on the
Prior Analytics covered the hypothetical syllogistic in considerably more detail [Maroth,
1989], but it is unlikely that it did. Avicenna almost certainly had read that commentary,
yet we find in The Cure that
... we came across a book on conditional (propositions and syllogisms) at¬
tributed to the most excellent among later (scholars). It seems to be wrongly
imputed to him. It is neither clear nor reliable. It neither gives an extensive
survey of the subject nor does it achieve its purpose. It gives a mistaken ex¬
position of conditional propositions, of a large number of syllogisms which
accompany them, of the reasons for productivity and sterility, and of the num¬
ber of moods in the figures. The student should not pay any attention to it—it
is distracting and misleading. ([Shehaby, 1973b, page 159]; cf. [Avicenna,
1964, page 356.10-15])
On coming across an alternative treatment of the hypothetical syllogistic, Avicenna thought
it was not Alfarabi’s, but he did not know. If Alfarabi had treated the hypothetical syl¬
logistic at any length in his Long Commentary, Avicenna would have known for sure
whether or not the attribution of the book to Alfarabi was correct. Later, Averroes would
have referred to Alfarabi’s longer treatment of the hypothetical syllogistic when treating
the problem in one of his essays (see below page 565). Avicenna did not know for sure,
Averroes did not refer to the longer treatment. We have already the main burden of what
Alfarabi wanted to present about the hypothetical syllogistic in developing his syllogistic.
Avicenna on the hypothetical syllogistic Alfarabi did not think Aristotle had written a
separate treatise on the hypothetical syllogistic, and he did not think that the hypothetical
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545
syllogistic merited extensive treatment. But he was aware of another, divergent tradition
on this point. Against Alfarabi’s comments in On Interpretation we must compare Avi¬
cenna’s comments in The Book of the Syllogism from The Cure, on the proof by reduction
and its proper analysis.
The only thing invoking this pointless exertion from people is the fact that
they have lost the work that Aristotle wrote detailing the hypothetical syllo¬
gistic. ([Avicenna, 1964, page 397.4-5]; cf. [Shehaby, 1973b, page 190])
Avicenna, that is to say, was part of the tradition which claimed extensive treatment of the
hypothetical syllogistic by Aristotle, a tradition known to Alfarabi when he was writing
his long commentary on On Interpretation.
This point makes a radical difference to how the two men write their treatises on logic.
Avicenna devoted substantial portions of The Cure to the hypothetical syllogistic (trans¬
lated in [Shehaby, 1973b]). He was clear that he got his treatment from elsewhere.
In our native country we came to know a long annotated book on this subject
which we have not seen since we left our country and travelled around to
look for a means of living. However, it might still be there. ([Avicenna,
1964, page 356.7-]; cf. [Shehaby, 1973b, page 159])
Two observations should be made at this point. Firstly, Avicenna did not create differences
from the Farabian system, but followed existing ones. Secondly, he was concerned to
modify the syllogistic so he could accommodate the doctrine of this non-Aristotelian text.
Avicenna on proof by reduction To repeat: there are many important aspects of Avi¬
cenna’s doctrines on hypothetical propositions and hypothetical syllogistic which cannot
be considered in this chapter. Here are three, which I mention because Averroes com¬
mented on them (see below page 566 f.). Firstly, Avicenna wrote at one point
All conditional and disjunctive propositions, and in particular the conditional
in which the antecedent and the consequent share one part, can be reduced
to categorical propositions—as when you say, for example, “If a straight line
falling on two straight lines makes the angles on the same side such and
such, the two straight lines are parallel.” This is equivalent in force to the
categorical proposition: “Every two straight lines on which another straight
line falls in a certain way are parallel.” ([Avicenna, 1964, page 256.11-15];
cf. [Shehaby, 1973b, page 55])
though his full doctrine on this matter is nuanced and complicated (see e.g. [Avicenna,
1964, page 264 f.]; cf. [Shehaby, 1973b, page 62]). Secondly, Avicenna held that a syllo¬
gism which conveyed new knowledge had to depend in the final analysis on a categorical
syllogism, which is therefore in this sense primary [Avicenna, 1964, page415 f.]. Thirdly,
Avicenna’s analysis of hypothetical syllogisms and categorical syllogisms includes claims
about the epistemic immediacy of the inferences [Avicenna, 1964, page 416.12 ff.]. All
these points deserve careful study, which they do not receive in this chapter.
546
Tony Street
What I do examine (and it has been examined before in [Shehaby, 1973b, page 277 f.])
is the way Avicenna accommodated the hypothetical syllogistic so that it goes to his analy¬
sis of proof by reduction. I present very briefly Avicenna’s placement of the hypothetical
syllogistic and its use in explaining the way Baroco is proved. I do so because it ex¬
emplifies how seamlessly Avicenna fitted extra-Aristotelian doctrine into his syllogistic.
To do this, I turn from Avicenna’s long exposition of the hypothetical syllogistic in The
Cure to the more managable exposition in Pointers [Avicenna, 1971 2 ; Goichon, 1951;
Inati, 1981], which is limited in its presentation to just those three parts of the hypotheti¬
cal syllogistic actually used in the explanation of Baroco [TusT, 1971, page 441],
As first studied in [Rescher, 1963d], Avicenna’s hypothetical propositions are quanti¬
fied. Always: when A is B, then D is H is an a-conditional; sometimes: when A is B, then
D is H is an i-conditional; never: when A is B, then D is H is an e-conditional; and some¬
times not: when A is B, then D is H is an o-conditional. They and categorical propositions
contribute to inferences, which Avicenna divided into conjunctive ( iqtirani ) and exceptive
( istitna’i ). This division is one of the points in his logic where he claimed for himself the
doctrine put forward:
According to what we ourselves have verified, syllogistic divides into two,
conjunctive ( iqtirani) and exceptive (istitna’i). The conjunctive is that in
which there occurs no explicit statement [in the premises] of the contradic¬
tory or affirmation of the proposition in which we have the conclusion; rather,
the conclusion is only there in potentiality, as in the example we have given.
As for the exceptive, it is that in which [the conclusion or its contradictory]
occurs explicitly [in the premises]. [Avicenna, 1971 2 , page 374]
Avicenna built his wholly hypothetical conjunctive syllogistic from quantified condition¬
als:
From the conditionals may be composed the three figures, just like the figures
of the categorical—they share in a consequent or an antecedent, and differ in
a consequent or an antecedent, just as the categoricals share in a subject or
a predicate, and differ in a subject or a predicate. The status [of one] is the
status [of the other], [Avicenna, 1971 2 , pages 435-436]
An example of such a syllogism would be (Barbara):
Always: when A is B then J is D, and
Always: when J is D then H is Z, which produces
Always: when A is B then H is Z
There are also a number of rules, including ecthesis, which deliver fourteen moods in the
wholly hypothetical syllogistic ([Avicenna, 1964, pages 295-304]; cf. [Shehaby, 1973b,
pages 91-99]).
Avicenna then considered the conditional premise conjoined with a categorical:
The conditional may be joined with a categorical. The most natural of these
[conjunctions] is when the categorical shares the consequent of the affirma¬
tive conditional, in one of the ways categoricals share [a term with each
Arabic Logic
547
other]. Then the conclusion will be a conditional whose antecedent will be
the very antecedent [of the first conditional], and whose consequent will be
the conclusion of the composition of the consequent conjoined with the cat¬
egorical. An example is: If A is B then every J is D, and Every D is H* it
follows that If A is B, then every J is H. It is up to you to enumerate the rest of
the divisions from what you have learned. {Avicenna, 1971 2 , pages 440-441 ]
Lastly, in a separate section, Avicenna listed the exceptive hypothetical syllogisms, which
include modus ponendo ponens, modus toliendo ponens, modus tollendo tollens, and
modus ponendo tollens [Avicenna, 1971 2 , pages 448-452],
With this material, we may follow how Avicenna analyses Baroco [Avicenna, 1971 2 ,
page 453 f.]. In schematized form:
To prove: Given all Bs are Ds and some Js are not Ds, then some Js are not
Bs.
1. Conjunctive:
When it is not the case that some Js are not Bs, then all Js are Bs. And all Bs
are Ds.
Therefore: when it is not the case that some Js are not Bs, then all Js are Ds.
2. Exceptive:
When it is not the case that some Js are not Bs, then all Js are Ds.
But it is not the case that all Js are Ds.
Therefore: It is not not the case that some Js are not Bs.
I would think that this could be extended programmatically to cover all proofs by re¬
duction, although Avicenna did not himself do so. In any event, Avicenna’s analysis failed
to impress the majority of logicians who followed him. The analysis allows us to observe
attitudes to Avicennan logic in the later tradition, however, as will become apparent (see
page 576 below). It is also significant, as noted above, because it shows just how Avicenna
fitted a non-Aristotelian tradition into his treatment of the categorical syllogistic.
3.5 Avicenna on Alfarabi on the modal logic
Avicenna and the tradition to which he belonged had a very different approach to the
hypothetical syllogistic from the one that Alfarabi and the Baghdad school had. There is
an even larger difference in the treatment of the categorical syllogistic, both as it is made
up of unmodalized and modalized propositions.
Differences concerning the absolute Basic to the many differences between Avicenna
and Alfarabi in treating the modal logic is their difference regarding the absolute propo¬
sition ( al-qadiyya al-mutlaqa), the ‘existential’ proposition ( al-qadiyya al-wugudiyya),
and their truth-conditions. Why these terms came to be used the way they were need not
concern us here (but see [Lameer, 1994, page 55 ff.], whence I draw the information in
this paragraph). In any event, mutlaqa was used in the mid-800s to render Aristotle’s ton
hyparcliein protasis as well as in connexion with the synonymous hyparchousa protasis.
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whereas wugudiyya seems to have been preferred later (about 900) to render hyparchousa.
Lameer’s results not only make clear the importance that the lexical preferences of the
translators have for coming to grips with the technical terms of the Arabic philosophical
tradition; they also make clear the fact that Alfarabi’s usage of both al-qadiyya al-mutlaqa
and al-qadiyya al-wugudiyya may be translated as ‘assertoric proposition’; and, finally,
that Alfarabi begins his logical work by constructing an assertoric syllogistic [Lameer,
1994, chapter four].
By contrast, Avicenna did not put forward an assertoric syllogistic and then modal-
ize it. Propositions 10 and 24 in appendix two serve roughly to show how Avicenna
took the absolute ( al-qadiyya al-mutlaqa ) and the ‘existential’ proposition ( al-qadiyya
al-wugudiyya\ henceforth referred to as the special absolute), in that their contradictories
are perpetuals or disjunctions of perpetuals (proposition 5 in appendix two). It is easy to
understand the way Avicenna understood the absolute proposition—his favourite example
of it is all men sleep. That is to say, an absolute a-proposition is taken as concealing an ‘at
least once’; all Frenchmen drink wine is not naturally taken to mean that they drink wine
constantly, but at least once in their lives. The unmodalised e-proposition by contrast
can be taken to convey perpetuity: no teetotaller drinks wine means that no teetotaller
ever drinks wine. Avicenna took the customary understanding of an e-proposition to be
perpetual, but stipulated that for logical purposes, it was to be taken as concealing the
same temporality as the a-proposition. The squares generated by the absolute and the per¬
petual are isomorphic with the squares generated by one-sided and two-sided possibility
taken with the necessary proposition (respectively, propositions 13, 26 and 1 in appendix
two), which are sometimes referred to as the classical squares of modal opposition IThom,
1996, pages 13 & 15].
Further, Avicenna rejected the conversion of the absolute e-proposition. Whereas con¬
tradiction for his simple propositions (dati; see below 550) depends on the modality or
temporality of the predicate, conversion depends further on the modality of the subject-
term. This is where we have the opportunity directly to compare what he was doing with
what Alfarabi was doing. Avicenna rejected the conversion by citing the counterexample,
no man is laughing. The only way to have the conversion go through according to him is
to take all men at the time or times that they are not laughing. One way then is to have
at time t: no men are laughing , which would convert as at time t: no laughing thing is
a man. Avicenna rejected this as a solution, because the men of a given time are not all
men, as should be the case in a proposition ready for logical treatment. (See further on
the conditions under which a proposition may be read at page 550 below.)
Proofs for e-conversion did not impress Avicenna. He had seen Alexander’s proof for
e-conversion as given in On the conversion of propositions, which runs as follows:
It may also be possible to prove conversion of the e-proposition by reduction
to the absurd. So if A belongs to no B, and B belongs to some A, these
combine by Ferio to mean that some A does not belong to A.* Since this is
impossible, its contradictory is necessary, which is that B belongs to no A.
[Alexander, 1971, page 65.6-9]
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549
Whether Avicenna thought Alexander was entitled to the proof is unclear, but Alfarabi
had adopted the proof, and Avicenna held it to be inconsistent with other doctrines he
held.
He who is eminent among the later scholars made this claim [for e-conversion]
with a good argument: [no Js are Bs converts to no Bs are Js,] if not, then
some Bs are Js; but no Js are Bs. This is a perfect syllogism [Ferio], self-
evidently productive. It is only made known afterwards by way of reminding
us, not to convey knowledge of which we are ignorant. From the above it fol¬
lows that some Bs are not Bs—this is absurd. [Avicenna, 1964, page 81.1-4]
The reason Alfarabi could not use, for example, the stratagem of the as-of-now proposi¬
tion to escape the counterexample was because he, like Avicenna, held that the subject-
term in a proposition ready for logical treatment could not be limited to those things that
fall under it at a given time (see next section, and also page 562 below).
On the other hand, if the subject is taken in the way chosen by the eminent
later scholar, such that J is whatever can be J, so that everything that can be
J, even if it exists or not or it is not the case that it is J, enters under it. Let
what follows from this be investigated... [Avicenna, 1964, page 85.5-7]
Differences concerning the modal logic Alfarabi’s ampliation of the subject-term got
him all the inferences in Aristotle, though often, as was pointed out later by Averroes,
with such obviousness that the proofs put forward by Aristotle become pointless (see page
562 below). One inference it gave Alfarabi was Barbara LXL, against the well-known
Theophrastean objection.
Know that the eminent scholar with whom I am most concerned to conduct
my discussion agrees with what I say; indeed, the First Teacher believes that
if the necessary major in the first figure is joined to a non-necessary minor,
the conclusion is necessary. Let us assume all Js are Bs non-necessarily, and
all Bs are As necessarily, and it yields what the eminent scholar and the First
Teacher both agree on, and what you have learned. [Avicenna, 1964, page
148.9-12]
Once again, however, Alfarabi is accused by Avicenna of holding incompatible doc¬
trines. Straight after noting that Alfarabi accepts Barbara LXL, Avicenna goes on:
But why doesn’t one of them go on to say that this is not a necessary propo¬
sition, but rather must be: all Bs are As necessarily in so far as they are Bs...
If this is taken into account, then what the detractors say against those who
produce a necessary from these premises turns out to be true. That is because
what the detractor is saying in this matter is like what the eminent scholar had
to say about the conversion of the possible proposition... [Avicenna, 1964,
page 148.13—pu]
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Tony Street
The ‘detractor’, a Theophrastean, would have as the conclusion all Js are As necessarily
in so far as they are Bs. Alfarabi is open to this objection because of the way he used
a reduplicative proposition to have the conversion of the two-sided possible go through.
His arguments on this point are reported by Avicenna (who by contrast held the two-sided
possible to convert as a one-sided possible [Avicenna, 1971 2 , page 340]) as follows:
That which a certain eminent scholar said is this: every animal is possibly
sleeping in so far as it is sleeping, so some of that which is sleeping is in
so far as it is sleeping possibly an animal, because its being an animal does
not belong to it in so far as it is sleeping—this is sheer sophistry. As for that
which rightly should be known about this matter, it is something the proof
for which has been given above. That which we ought to repeat and set down
here is that the utterance ‘in so far as it is sleeping’ is said either as part of the
predicate, or as part of the subject. If it is part of the predicate, then it must
first off in conversion be made part of the subject, thus: Some of that which
is sleeping in so far as it is sleeping is possibly an animal; this is as you hear
it [that is, it is gibberish]. Given that it is true, it is not what we are talking
about...
But you know, O eminent one, that the sleeping taken without condition is
other than the sleeping taken with the condition of its being sleeping, and in
so far as it is sleeping... [Avicenna, 1964, pages 209.7-210.5]
Avicenna, that is to say, took Alfarabi to task for using one solution to get out of the
problem of the conversion of the two-sided possible, but not continuing to bind himself
by that solution in solving the problem of the two Barbaras. Avicenna placed both stricter
and looser demands on logical exegesis: it need not follow Aristotle everywhere, but it
must be internally consistent. More generally, Alfarabi was trying to find a way to make
sense of the Aristotelian text, proposing solutions to local problems, and hoping that the
ideas behind the text would ultimately shine through, whereas Avicenna did not think that
there was any point in trying to give a literal exegesis of Aristotle’s syllogistic.
You should realize that most of what Aristotle’s writings have to say about
the modal mixes are tests, and are not genuine opinions—this will become
clear to you in a number of places... [Avicenna, 1964, page 204.10-12]
Avicenna’s modal syllogistic So how did Avicenna build his modal syllogistic? The
first important feature to note is that having taken the absolute as a temporal, he placed it
within his syllogistic alongside the modals.
The second important feature in Avicenna’s syllogistic is the conditions under which
a proposition can be read. There is no distinction made in Arabic logic corresponding to
the Western distinction between divided and composite readings. The distinctions Avi¬
cenna proposed, however, became just as important and pervasive for logicians writing
in Arabic. There are four intrinsic and two extrinsic conditions under which propositions
can be read. (Although this passage is given for a necessary proposition, these conditions
are applied to propositions with other modal operators.)
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551
Necessity may be (1) absolute, as in God exists ; or it may be connected to
a condition. The condition may be (2) perpetual for the existence of the
substance ( dat ), as in man is necessarily a rational body. By this we do
not mean to say that man has been and always will be a rational body, be¬
cause that would be false for each given man. Rather, we mean to assert
that while he exists as a substance (ma dama mawguda d-dat ), as a human,
he is a rational bodyL Or the condition may be (3) perpetual for the sub¬
ject’s being described in the way it is ( dawcima kawni l-mawdu‘i mawsiifan
bi-nia wudi'a ma'ahu), as in all mobile things are changing ; this is not to be
taken to assert that this is the case absolutely, nor for the time [the subject]
exists as a substance, but rather while the substance of the moving thing is
moving. Distinguish between this condition and the first condition, because
the first has set down as the condition the principle of the substance, ‘man’,
whereas here the substance is set down with a description (sifa) that attaches
to the substance, ‘moving thing’. ‘Moving thing’ involves a substance ( dat
wa-gawhar) to which movement and non-movement attach; but ‘man’ and
‘black’ are not like that.
Or it may be a condition (4) of the predicate; or (5) of a definite time, as in
an eclipse; or (6) of an indefinite time, as in breathing. [Avicenna, 1971 2 ,
264-266]
This passage draws on earlier Peripatetic writings (for an analysis of its probable sources,
see [Back, 1992]), and it is best understood as the way Avicenna laid out various modal
notions. Avicenna’s interests were, with one exception, exclusively in propositions read
under the second condition (the dati, or substantial reading) and the third (the wasfi, or
descriptional reading), but later Avicennan logicians also investigated the fifth (the waqti,
or temporal) and the sixth (the muntashir , or spread). (See the renditions in appendix two
at page 592 below; £ renders the dati, C the wasfi, and T and S the fifth and sixth waqti
readings.)
I think that the dati reading, although it turns on a distinction different from the one
which delivers Abelard’s divided reading, is functionally the same as the divided. Two
examples may help clarify the distinction between it and the wasfi. All bachelors are
necessarily unmarried is true as a wasfi, because ‘bachelors’ picks out men just while they
are unmarried: all men while bachelors are necessarily unmarried. As a dati, however, it
is false: all bachelors are men, and it is untrue that all men are necessarily unmarried. By
contrast, (and this is the most common Avicennan example) all who sleep wake is true as
a dati (because every animal that sleeps also wakes up from time to time), but false as a
wasfi (because nothing can be awake while sleeping). No As are Bs while As, Avicenna
claimed in Pointers, would convert as no Bs are As while Bs, and would contradict some As
are Bs while As\ and would save the second figure for the account in the Prior Analytics.
Avicenna took great pride in the fact that his two readings of the propositions allowed him
to square a set of examples where other logicians had failed (see also page 578 below).
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The people who went before us were not able to reconcile us to their view
by their examples and usage. The explanation of this is lengthy. [Avicenna,
1971 2 , page 314]
Avicenna later in his work also investigated the way a wasfi major and a datT minor func¬
tion in a syllogism, but those investigations deserve extended study, and are beyond the
scope of this chapter (though see now [Thom, ]). Later logicians challenged Avicenna’s
claims for the way the wasfi contributes to an inference (see below page 575 f.), and devel¬
oped his insights extensively; even later, they included his extrinsic temporal conditions
in their investigations.
And the syllogistic with purely datT premises? Avicenna developed his datT modal
syllogistic as two isomorphic systems, one using temporal propositions (functioning in
contradiction and conversion like propositions 5,10 and 24 in the appendix) and the other
using modal propositions, which function in contradiction and conversion like these (the
numbers indicate the propositions in appendix two which replace Avicenna’s modals in
later logical writings):
1.* (Va;)[OA x D UB X )
13.* (Vtc)[OAz D OB x ]
26.* (Vx){OA x D [OB x kO~ B x )}
But Avicenna wanted the two sub-systems to interract in ways that show that these ren¬
ditions are not right—for example, Avicenna wanted syllogisms with possible minors, in
particular Barbara XMM, yet argued that absolute a- and i-propositions convert as ab¬
solute propositions. The system deserves serious study (for a description of the whole
system with datT premises, see [Street, ]).
It is important to note that Avicenna, though using the perpetual to provide a contradic¬
tory for the absolute, did not investigate how the perpetual contributes to other inferences.
I think that Avicenna wanted to provide a syllogistic that looks like it is treating only the
propositions that Aristotle examined. Later logicians were far less concerned with pre¬
serving that sort of contact with the Aristotelian tradition, and investigated the perpetual
as a fully-fledged member of their set of propositions. Working out Avicenna’s system is
the first and major problem for the study of medieval Arabic logic. It may be complicated
by non-logical factors: Avicenna saw himself as a second Aristotle, and it may well be
that certain features of his own logic are tests set to puzzle his readers.
3.6 Baghdad and the East
The history of post-Avicennan logic is the history of the eventual conquest of a system de¬
rived from Avicenna’s system over the logic taught in the Baghdad school. In this section
I offer a few reflections on the respective provenance and strengths of each tradition.
Firstly, I think that there are some grounds to believe that Avicenna was not offering
an entirely new system to which people had to be converted, but was merely setting in
sharper format a system which was already broadly accepted in Khurasan. At least on
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553
my reading of Pointers , Avicenna only claimed as his personal contributions to the formal
logic he presented the following: the coinage of the term wugudiyya, the use of the wasfi
as a plausible way to save Aristotle’s position on the absolute, and the division of the
syllogistic into conjunctive and exceptive [Avicenna, 1971 2 , respectively pages 309, 314,
374]. The last two contributions ramify through the rest of the system. Even without
them, however, the fact remains that the absolute would be treated quite differently from
the way Alfarabi had treated it, and attention would be paid to a body of hypothetical
syllogistic ignored by the Baghdad philosophers. This is at least compatible with the
conclusion that the logic studied in Khurasan was quite different from Baghdad logic
before Avicenna arrived. Avicenna came to an Aristotle mired in nearly one and a half
millenia of interpretations, and the specificities of the tradition in Khurasan may have been
paramount in determining what he did with Aristotle, and perhaps also in determining
what those who came after him did. This speculation is not meant to deny that Avicenna’s
formulation of that logic was the strictest, and the one to which subsequent logicians, both
friendly and unfriendly to the project, had first recourse.
Other reasons for the wide acceptance of the Avicennan tradition of logic have to do
with the general fortunes enjoyed by the larger philosophical system Avicenna put for¬
ward. That system, having been presented at many points in Avicenna's writings as
congenial with Islam, proved to be so adaptable to the needs of Islamic philosophical
theology that by the beginning of the twelfth century, people understood by ‘philosophy'
Avicenna’s synthesis. Even more generally, although Avicenna was born into the dying
days of one dynasty (the Samanids), he moved and worked through the halcyon realms of
the dominant force of the era, the Buyids. Baghdad had lost much of the political and cul¬
tural prestige it had enjoyed in the tenth century, and the dynamics of political hegemony
in the Islamic world were driving from the East through to the West.
Finally, it must be borne in mind that although places like Khurasan and other eastern
realms quickly became almost wholly Avicennan, what that means is much more complex
than appears on the face of it. None of these logicians, so far as I am aware, adopted the
Avicennan syllogistic in its entirety, though most adopted Avicenna’s three most charac¬
teristic doctrines. So these logicians nearly all agreed that the syllogistic divides broadly
into conjunctive and exceptive; they further agreed that the hypothetical syllogistic is im¬
portant, and nearly all devoted analyses to aspects of the hypothetical syllogistic. That
said, they did not necessarily agree on precisely what it is that matters most about the hy¬
pothetical syllogistic, or how exactly it relates to the categorical syllogistic. Secondly, the
logicians in these regions nearly all delivered a syllogistic system that uses an Avicennan
absolute proposition. Thirdly, they all investigated propositions read under some or all of
Avicenna’s conditions. This is the sense in which these logicians are Avicennan.
Baghdad logic, by contrast, did not prosper. One thing that did not limit its influence
was the fact that it was a Christian school in an Islamic society, constituted mainly by
Christians, with only a few Muslims such as Alfarabi and Abu-Sulayman as-SigistanT
among its members. Avicenna’s disciples included a Zoroastrian and a Christian; philos¬
ophy throughout the Islamic world tended to be accepted, by those people who accepted it
at all, as a discipline which would attract people from various faith-communities. Again,
it was not internal dissension that weakened Baghdad; in spite of the sectarian differences
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between the various Syriac confessions, it appears that in Baghdad a collegial and cordial
spirit prevailed [Zimmermann, 1981, page cxii]. In fact, Christian and Muslim logicians
faced parallel opposition within their respective faith-communities, and the apologia for
logic was a genre they were both forced to write (for a Christian example, see [Rescher,
1963f]).
Baghdad was unlucky in the successors it had to Alfarabi. Yahya ibn-‘AdT wrote ex¬
tensively and competently on logic, although he apparently argued that modal logic was
ill-conceived [Endress, 1977, chapter 3, and page 51]. The only Baghdad philosopher
Avicenna admired was Alfarabi; the rest of the Syriac Christians he dismissed as wooden-
minded in logic (see above page 535). He famously and witheringly said of Ibn-at-Tayyib,
his contemporary and head of school in Baghdad, that his work was best sent back to the
bookseller, whether or not a refund was offered [Gutas, 1988, page 68], Alfarabi had been
a continuation of the Syriac tradition, but Ibn-at-Tayyib was a mere replication of it. (Of
course, this is merely to repeat Avicenna’s judgement and, as Lameer reminds us, we have
yet properly to check Ibn-at-Tayyib’s writings; see [Lameer, 1996, page 96].) Still, for all
the undoubted decline in its philosophical fortunes, Baghdad still produced a considerable
logical posterity. The Andalusian Muhammad ibn-‘Abdun (d. c. 995) came to Baghdad to
study with Abu-Sulayman, whereupon he went back to Spain and inaugurated a tradition
of logicians who can only be understood against the tradition of Alfarabi. In fact, what
modal logic we can guess was being taught in Baghdad was probably the modal logic we
find in early Spain (see below pages 561 & 567).
For the most part, the Baghdad school continued to concentrate on a range of tasks
among which exegesis, or really, summary, figured prominently. If Galen’s logic was still
read at all, it was here in Baghdad—Avicenna had followed Alexander, and dismissed
him as “the man who was strong in medicine but weak in logic” ([Shehaby, 1973b, page
5]). But even in Baghdad teaching changed after Avicenna. Avicenna’s philosophy was
the most significant intellectual challenge the Baghdad philosophers had to face, and even
during Avicenna’s lifetime, they tried to meet that challenge. After Avicenna a new strain
is apparent in Baghdad logic. References and reflexes can be found in writings from
Baghdad in which the Avicennan position is set down and then dismissed. This apolo¬
getic becomes a new theme in the Baghdad school, and subsequently in Spain. In the
end, however, even among the Syriac Christians, Avicenna’s system prevailed (see e.g.
[Jannssens, 1937]).
4 LOGIC AND THE ISLAMIC DISCIPLINES
The Islamic disciplines include law and jurisprudence, Koranic exegesis, analysis of tra¬
ditions relating to the Prophet, theology and grammar. Together these disciplines func¬
tion to determine the Islamicity of the spiritual and public life of a society. The central
doctrines and techniques of these disciplines reached what later came to be considered
their classical formulation by the end of the second Abbasid century, which was about
the same time that a truly naturalized Arabic logic was being achieved. People began to
ask whether logic was a discipline constituted in ways similar to the Islamic disciplines,
whether it was useful for the Islamic disciplines, whether, indeed, it was even compatible
Arabic Logic
555
with them. Some of Alfarabi's logical writings are attempts to answer these questions:
he linked logic into an ancient tradition, and showed great concern to make its technical
terms perspicuous to speakers of pure Arabic, both important matters in the constitution
of an Islamic discipline; he was concerned to make the forensic utility of logic obvious;
and he stressed its parallels to grammar (see above page 536 ff.).
The single most important voice in the arguments over the centuries about logic and its
relation to the Islamic disciplines is that of the famous jurist and theologian, Abu-Hamid
al-Gazall (d. 1111). Some fifty or so years after Avicenna’s death, GazalT argued that
logic was both licit and useful for theology and jurisprudence. In this he was following
the example of Alfarabi. The clarity of GazalT’s prose style and the depth of his spiritual
insights have won him enormous prestige in the Islamic community. His arguments that
there is nothing inimical to religious belief in logical studies derived much of their force
from that prestige. He did not end the attacks on logic, but his arguments in support of
logic are probably the most decisive factor in its inclusion as a subject for study in the
madrasa.
Logic and grammar What sort of resistance did logic face? The most frequently cited
example of the clash between the Islamic disciplines and logic is the debate in Baghdad
conducted between Abu-SaTd as-STraff (d. 978) and Alfarabi’s senior colleague, Abu-
Bisr Matta (d. 940). The study of this debate is now a something of a minor industry
within the field of Islamic studies, and it is undoubtedly important for revealing how logic
was received among some of the educated classes of Baghdad at the end of the translation
movement described in section 2 above. The debate was convened by the vizier, who
invited Abu-Bisr to defend the value of Aristotelian logic relative to that of the Arabic
grammatical tradition. Abu-Sa‘Td, a young but promising grammarian, stepped forward
to put the case for grammar, and won the debate by acclaim, humiliating Abu-Bisr in
the process. It should be said, however, that important as the debate may have been, and
much studied as it is, there continues to be scholarly disagreement as to what point it is
that Abu-Sa‘Td was trying to make. One account of Abu-Sa'Td’s attack has him defending
the ambient Platonism of Baghdad against the rising peripateticism of Abu-Bisr and his
colleagues [Mahdi, 1970]; more commonly, he is taken to represent the practitioners of
the Islamic disciplines and their worries about the far-reaching claims made for logic
([Elamrani-Jamal, 1983, pages 61-71]; briefly in [Arnaldez, I960-], at length in [Endress,
1986]). It has also been pointed out that we may be trying to extract more from the debate
than the occasion of its convention (an amusement for the vizier) allows [Frank, 1991];
whatever the specific points in Abu-Sa‘Td’s arguments, one is left with the impression that
logic is not so much dangerous as merely laughable.
In the debate, we find Abu-Sa‘Td again and again chiding Abu-Bisr for his bad Arabic,
and for his naive confidence that knowledge of logic can somehow protect him from error
in thinking and end dispute in philosophy.
The world remains after Aristotle’s logic as it was before his logic. Resign
yourself, therefore, to dispense with the unattainable, since such a thing is
wanting in the creation and nature of things. If, therefore, you were to empty
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your minds of other things, and devote your attention to the study of the
language in which you are conversing and disputing with us, and instruct
your friends in words which the speakers of that language can understand,
and interpret the books of the Greeks in the style of those who know the
language, you would learn that you can dispense with the ideas of the Greeks
as well as you can dispense with the language of the Greeks. [Margoliouth,
1905, pages 115-116]
Abu-Sa‘Td’s most stinging taunts were directed against the technical jargon that Abu-
Bisr and his colleagues were using, which Abu-Sa‘Td showed to be more a hindrance
than a help to clear thinking. He also mocked the claims the philosophers made for the
importance and utility of logic. Among other things, he cast doubt on the coherence of
their claims that there is a higher grammar, common to all languages.
Logic and theology The theologians of tenth-century Baghdad also studied argument-
forms, and had an elaborate set of terms to classify arguments as good or bad. These
terms, however, map so precisely onto Stoic terms and function in such a similar way that
it has been concluded that theological logic almost certainly derives from Stoic logic [van
Ess, 1970], That said, the process by which Stoic logic came to Baghdad is far from clear,
and it is only on the grounds of structural and terminological similarity that the claim
can be made. (To get an idea of how speculative these assessments of Stoic origin are,
see [Shehaby, 1973a] and especially the discussion following it; see now [Gutas, 1994].)
Whatever their origin, theologians had methods to evaluate arguments which were not
Aristotelian.
Still, it is one thing to have a system for evaluating arguments, but quite another to say
that no other system should be studied; yet that is what some theologians did argue. We
can get some idea of why they did so from the great fourteenth-century intellectual histo¬
rian, Ibn-Haldun (d. 1406). Ibn-Haldun had himself written a short treatise on logic in his
younger days, and there are many references to logic running through his Prolegomena
[Ibn-Haldun, 1958]. He was, in short, an interested and quite probably competent witness
to the state and history of logic in his time. The passage he wrote devoted solely to logic
divides into two sub-histories, one on the Organon in the Islamic world, and the other on
the tensions between the logicians and the theologians.
It should be known that the early Muslims and the early speculative theolo¬
gians greatly disapproved of the study of this discipline. They vehemently
attacked it and warned against it. They forbade the study and teaching of
it. Later on, ever since GazalT [d. 1111] and Fahra ddTn ar-RazT [d. 1210],
scholars have been somewhat more lenient in this respect. Since that time,
they have gone on studying logic, except for a few who have recourse to the
opinion of the ancients concerning it and shun it and vehemently disapprove
of it. Let us explain on what the acceptance or rejection of logic depends, so
that it will be known what scholars have in mind with their different opin¬
ions... ([Ibn-Haldun, 1858, page 113.13—u]; cf. [Ibn-Haldun, 1958, pages
143-144])
Arabic Logic
557
Ibn-Haldun went on to give a short history of Islamic theology, the arguments that it
developed in defence of the articles of faith, and the atomistic and nominalist metaphysics
which was simultaneously refined to support those arguments, that is to say, classical
kalam atomism.
It then came to be the opinion of As’arT, BaqillanT, and Isfara’TnT [famous
exponents of the classical kalam], that the evidence for the articles of faith
is reversible in the sense that if the arguments are wrong, the things proven
by them are wrong. Therefore, BaqillanT thought that the arguments for the
articles of faith hold the same position as the articles of faith themselves and
that an attack against them is an attack against the articles of faith, because
they rest on those arguments. ([Ibn-Haldun, 1858, page 114.13-16]; cf. [Ibn-
Haldun, 1958, pages 144-145])
But Aristotelian logic, Ibn-Haldun went on, assumes the five universals and the common¬
places for topical reasoning, and assumes further that they have an extramental existence.
This assumption is incompatible with the theologians’ denial that universals have a real
existence. If the theologians are right, then
... all the pillars of logic are destroyed. On the other hand, if we affirm their
existence, as is done in logic, we thereby declare wrong many of the premises
of the theologians. This, then, leads to considering wrong their arguments for
the articles of the faith, as has been mentioned before. This is why the early
theologians vehemently disapproved of the study of logic and considered it
innovation or unbelief, depending on the particular argument declared wrong
by the use of logic.
However, recent theologians since GazalT have disapproved of the idea of the
reversibility of arguments and have not assumed that the fact that the argu¬
ments are wrong requires as its necessary consequence that the thing proven
by them be wrong. They considered correct the opinion of logicians concern¬
ing intellectual combination and the outside existence of natural quiddities
and their universals. Therefore, they decided that logic is not in contradic¬
tion with the articles of faith, even though it is in contradiction to some of
the arguments for them. In fact, they concluded that many of the premises
of the speculative theologians [who followed classical kalam] were wrong.
For instance, they deny the existence of atomic matter and the vacuum and
affirm the persistence of accidents and so on. For the arguments of the theolo¬
gians for the articles of the faith, they substituted other arguments which they
proved to be correct by means of speculation and syllogistic reasoning. They
hold that this goes in no way against the orthodox articles of faith. This is
the opinion of RazT, GazalT, and their contemporary followers. ([Ibn-Haldun,
1858, page 115.13-116.8]; cf. [Ibn-Haldun, 1958, pages 146-147])
In other words, Gazali and Razi were worried at equating the credibility of their faith
with the credibility of kalam atomist theory. It is the theological decision to overturn
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the theory of the reversibility of arguments which found space for logical studies in a
theological education.
That at least is Ibn-Haldun’s account. It suffers from some problems which cannot be
considered here (though compare the account in [Marmura, 1975]). In any event, GazalT
not only argued in defence of logic, and used it in his theological works; he went further
and argued in The Just Balance [GazalT, 1959] (translated [Brewster, 1978; McCarthy,
1980], and studied [Kleinknecht, 1972]) that the Koran sanctioned its use, if in slightly
coded language. But the fact that theologians condoned it may not have been the deciding
factor in the acceptance of logic.
Logic and law The law is sometimes said to be the most important of the Islamic disci¬
plines. Lawyers had been reflective about their system of reasoning, and analogy (which
they called qiyas, the same word used by the logicians for ‘syllogism’) as well as other
techniques were the centre of a series of sophisticated discussions. GazalT was first and
foremost a lawyer, and held his chair in Safi‘I jurisprudence at the Nizamiyya in Bagh¬
dad from 1091. By showing legal arguments ultimately to depend on the syllogism, and
by prefacing his last juridical summa, The Distillation of the Principles of Jurisprudence
[Gazall, 1938], with a logical treatise, GazalT did more than any other earlier scholar to
have logic made part of madrasa studies.
In Distillation, GazalT referred to two of his earlier works on logic, The Touchstone
for Speculation [GazalT, 1966] and The Yardstick of Knowledge [GazalT, 1961b], both of
which he wrote after writing his famous Intentions of the Philosophers [GazalT, 1961a],
Intentions is in fact a pretty close Arabic paraphrase of Avicenna’s Persian Philosophy for
‘Ala'addawla, which contains a very elementary treatment of logic (English translation,
[Avicenna, 1971]). In assessing GazalT as a logician it must be said that, from a formal
point of view, he never rises above Intentions.
The first section of Touchstone follows the structure of Intentions, but in the second
and third sections, GazalT’s interest in cognitive aspects of premises, in the pragmatics
of argument, and in legal problems, comes to the fore. Throughout the book, there is a
concern to find new ways of putting the philosophers’ terms of art, ways that correspond
to terms used in the Islamic disciplines (though GazalT was careful to point out when there
are differences between logical and grammatical usage). In Touchstone , GazalT had more
fully than any other logician up to that time gone into the problems of naturalising logical
terminology; and he had more comprehensively shown how it can contribute to the prag¬
matic needs of juristic reasoning. At the end of Touchstone, GazalT advised his readers to
go to his Yardstick of Knowledge for fuller treatment of the material covered. Yardstick
does indeed give a much fuller exposition of the subject, with all the technical terms nor¬
mally used by the philosophers. None the less, the goals of Yardstick are identical with
those of Touchstone:
We shall make known to you that speculation in juristic matters ( al-fiqhiyydt )
is not distinct from speculation in philosophical matters ( al-'aqliyyat ) in
terms of its composition, conditions, or measures, but only in terms of where
it takes its premises from. [GazalT, 1961b, page 28.2^1]
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The questions that were important for Avicenna in his reading of Alfarabi, and which
came to be important generally for the major logicians of the post-Avicennan tradition (as
examined in section 5 below), did not matter at all for GazalT. Even the more advanced
Yardstick, though mentioning the distinction between dati , wasfi and temporal ( waqtl )
readings in propositions, and the modalities [GazalT, 1961b, pages 88-90], never con¬
siders how they contribute to an inference. There is never a doubt raised about whether
the unmodalised proposition (the only kind GazalT considered) will function like the as-
sertoric in the early books of the Prior Analytics. The relation between the categorical
and hypothetical syllogistic is treated insouciantly (the categorical “is sometimes called
an iqtirdm syllogism, sometimes a gazmT' [GazalT, 1961b, page 98.14-apu]), the hypo¬
thetical syllogistic is exemplified only by unanalysed propositions [GazalT, 1961b, pages
111-114], and the deduction involving a contradiction is treated without any reference to
the hypothetical syllogistic [GazalT, 1961b, page 114].
I think a few conclusions may be drawn from these considerations. For all Ibn-Haldun
says that theological reasons made logic acceptable to GazalT, juridical considerations
seem more significant. GazalT’s contribution to logic was mainly on the level of defus¬
ing objections to its study by domesticating its jargon, and showing by legal examples
its utility [Hallaq, 1990, page 315]. Although GazalT is sometimes said to be Avicennan
[Rescher, 1964, page 49], this is true only in an attenuated sense. Even though his trea¬
tises derive their logical content from Philosophy for ' Ala’addawla , his significant work
was done in the spirit of Alfarabi’s apologetics. He is in this sense more Farabian than
Avicennan. Further, his work was not paradigmatic for later theologians. Other theolo¬
gians began to study logic from the twelfth century onwards, but they did not all study it
in the same way, or for the same purposes. This is particularly clear from a comparison
of GazalT’s interests with those of RazT (see below page 572 ff.). GazalT was a promoter
of logic, but not a practitioner. Other, later theologians were often both.
Continued oposition to logic GazalT’s achievement was not the end of all opposition to
logic among Muslim scholars, though future attacks on logic never seriously affected the
study of the discipline. We find pious opposition taking a number of forms. One famous
example is a legal opinion issued by Ibn-as-Salah (d. 1245) on the reprehensibility of
logical studies.
As far as logic is concerned, it is a means of access to philosophy. Now
the access to something bad is also bad. Preoccupation with the study and
teaching of logic has not been permitted by the law-giver, nor has it been
suggested by his Companions or the generation that followed him, nor by
the learned imams, the pious ancestors, nor by the leaders or pillars of the
Islamic community whose example is followed. God has protected them
from its danger and its filth, and cleansed them of its uncleanness. The use
of the terminology of logic in the investigation of religious law is despicable
and one of the recently introduced follies. Thank God, the laws of religion
are not in need of logic. Everything a logician says about definition and
apodictic proof is complete nonsense. God has made it dispensable for those
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who have common sense, and it is even more dispensible for the specialists in
the speculative branches of jurisprudence. [Goldziher, 1981, pages 205-206]
Perhaps the most famous opponent of logic is the fideist jurist, Ibn-Taymiyya (d. 1328),
who was writing near the end of the period covered in this chapter. Ibn-Taymiyya wrote
a lengthy condemnation of the use of logic, Refutation of the logicians, which in a later
abridged form has been translated into English [Hallaq, 1993]. Logic is merely distracting
where sound intuition can hit the mark; so much for the claims made about the univer¬
sal utility of logic. On the more specific claim that syllogistic reasoning is valuable for
jurisprudence, Ibn-Taymiyya has nothing but derision. In fact,
their distinction between a categorical syllogism and analogy—that the for¬
mer is capable of leading to certainty while the latter to nothing but probability—
is invalid. In fact, whenever one of them leads to certainty so does the other,
and whenever one of them leads to nothing but probability the other does
likewise. When the evidence results in certainty or in probability, this is so
not because its form is syllogistic and not analogical, but rather because the
syllogism contains conclusive evidence. If either an analogy or a syllogism
encompasses a matter that entails a certain judgement, then certainty is at¬
tained. [Hallaq, 1993, page 125]
For all his contempt for logic, Ibn-Taymiyya never impugned the formal aspects of the
syllogistic, which had come by his day to be the major focus of the logical treatise. But
what is the value of this formal study?
The validity of the form of the syllogism is irrefutable ... But it must be
maintained that the numerous figures they have elaborated and the condi¬
tions they have stipulated for their validity are useless, tedious, and prolix.
These resemble the flesh of a camel found on the summit of a mountain; the
mountain is not easy to climb, nor the flesh plump enough to make it worth
the hauling. [Hallaq, 1993, page 141]
5 LOGIC AFTER AVICENNA
As has been noted (see page 552 f. above), Avicenna came to exercise an extraordinary
influence over subsequent generations of philosophers. For many logicians, their work
has to be understood as an attempt to extend or modify the Avicennan system. These
logicians no longer referred to the Prior Analytics as they went about their tasks, but to
Pointers and Reminders. Again, as has been noted, I refer to a logician as ‘Avicennan’
if he adopted the three central modifications Avicenna introduced into the formal syllo¬
gistic: the Avicennan truth-conditions for the absolute proposition; the readings under
which Avicenna read modal and temporal propositions; the division of the syllogistic into
conjunctive and exceptive syllogisms—though this way of deciding whether or not a lo¬
gician is Avicennan has the consequence of making quite a few logicians Avicennan who
none the less direct trenchant criticism against Avicenna. The Avicennan logicians con¬
trast most starkly with those logicians for whom the primary task was the recovery of a
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true Aristotelianism; Averroes was the major, though not unique, representative of this
tradition. But by the time Averroes came to grapple with the Aristotelian logical texts,
the Avicennan system was so dominant that the points to which Averroes had to direct
most of his exegetical energies had been determined for him by that system. Averroes
was something of an Avicennan in spite of himself.
There are other approaches to the writing of logical treatises in this period which are
less easily categorised. Some of the Syriac Christians continued to write logical com¬
mentaries much as they always had, seemingly with little or no reference to Alfarabi or
Avicenna. Other logicians in Baghdad in the twelfth century certainly had access to Avi¬
cenna’s writings, and Alfarabi’s manuscripts may well still have been available in Bagh¬
dad in the twelfth and thirteenth centuries. Given the few and stereotypical references
to Alfarabi among the later logicians working in Iran and further east, however, we must
wonder whether his manuscripts were available there. A tradition of logical studies in
Spain directed much of its attention to Alfarabi’s writings, and perhaps had less access
to the Avicennan texts. By the end of the twelfth century, the only traditions that really
mattered were the Averroist and the Avicennan. But by the end of the thirteenth century,
only the Avicennan mattered.
5.1 Spain and the Averroist project
Early logical studies in Spain Logic was first taught in Muslim Spain, so the biobibli¬
ographers would have it, when a Andalusian, Muhammad ibn-’Abdun (d. c. 990), studied
in Baghdad under Abu-Sulayman as-SigistanT, and then returned to his homeland to start
teaching the subject there (see generally [Dunlop, 1955]). lbn-‘Abdun was among the
teachers of Abu-‘Abdallah al-Kattanl (d. 1029) who in turn was one of the teachers in
logic of Ibn-Hazm (d. 1064), a man more famous for his work in poetry, jurisprudence
and theology, than for his logic. Still, his book, An approach and introduction to logic
(at-Taqrlb li-hududal-mantiq wa-madhaluhu), is interesting because it is another effort to
make Aristotelian logic acceptable to Muslim jurists (for a summary, see [Chejne, 1984]).
This is very like the project of GazalT, and of Alfarabi. But Ibn-Hazm was, like GazalT,
more a religious scholar than a logician, and his contemporaries and immediate successors
tended to belittle his logical achievements. Ibn-'Abdun was also among the teachers of
the teachers of Ibn-Bagga (d. 1138) and Averroes (d. 1198), an altogether more glorious
line of logicians.
Before turning to Ibn-Bagga and more particularly Averroes, it is interesting to note a
treatise we have by one of Ibn-Bagga’s contemporaries, Abu-s-Salt of Denia (d. 1134).
Abu-s-Salt's presentation of the Aristotelian modal system in his Setting minds straight
has been edited by Gonzalez Palencia and its results noted by Rescher [Gonzalez Pa-
lencia, 1915; Rescher, 1963a]. Abu-s-Salt stated the assertoric syllogistic [Gonzalez Pa¬
lencia, 1915, pages 20-29], then developed his modal syllogistic by setting down ex¬
actly the same moods and mixes we find in Aristotle, in exactly the ordering of the Prior
Analytics [Gonzalez Palencia, 1915, pages 29-46] (with two exceptions, being uniform
necessity moods and mixed necessity and problematic moods; but Abu-s-Salt explained
how to get an Aristotelian conclusion for each). Abu-s-Salt’s treatment of conversion
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[Gonzalez Palencia, 1915, page 21] betrays no concern for the Avicennan counterexam¬
ples. He devoted two pages to the hypothetical syllogistic, giving it only with unanalysed
propositions [Gonzalez Palencia, 1915, page 46-47]; and he did not call on it to explain
the workings of a deduction involving a contradiction [Gonzalez Palencia, 1915, page
48]. I think his treatise gives us an opportunity to see the standard treatment of the later
Baghdad school, and of the pre-Averroist school in Spain. It is nothing more than a set of
notes summarizing early parts of the Prior Analytics.
Ibn-Bagga was regarded by no less than Ibn-Haldun as a philosopher of the calibre of
Avicenna, Alfarabi and Averroes. He was seen by later scholars as inaugurating a new
and more rigorous era of logical studies in Spain. I do not think that the present state of
the field is such that we are able to judge Ibn-Bagga’s logical writings. None the less, he
obviously consecrated a great deal of effort to writing commentaries on Alfarabi’s logical
works (see for example [Alfarabi, 1986a]; a brief summary is given [Gutas, 1993, pages
54-55]). In this he prepared the ground for Averroes’ early logical training.
Averroes and the logical tradition It is beyond dispute that the major logician writing
in Muslim Spain was Averroes. Averroes has a vast output. Some of his work was directed
to the familiar task of showing the study of logic to be not only licit but actually incumbent
on Muslims [Hourani, 1961, pages 44-47]. Specifically on the Prior Analytics, we have a
middle length commentary and a collection. The Essays [Averroes, 1983b], which address
specific problems in the Aristotelian tradition. (For his logical writings, see [Gutas, 1993,
pages 55-56].) The Essays are very focused, and what follows derives from them.
The main point I hope to emerge here is what it means to say that Averroes wrote in the
Farabian tradition. It is much harder to consider Averroist logic (and Averroist philosophy
generally) than Avicennan logic, because Averroes was constantly revising his system.
Throughout his career, and even more ardently at the end, Averroes was trying to preserve
the insights of Aristotle. In the Essay on the modalities of conclusions following from the
modalities of premises [Averroes, 1983b, pages 176-186], Averroes wrote:
These are all the doubts in this matter. They kept occurring to us even when
we used to go along in this matter with our colleagues, in interpretations by
virtue of which no solution to these doubts is clear. This has led me now
(given my high opinion of Aristotle, and my belief that his theorization is
better than that of all other people) to scrutinize this question seriously and
with great effort. [Averroes, 1983b, page 181.6-10]
As a corollary of this constant revision, Averroes was changing his opinion and assess¬
ment of Alfarabi. This can obscure the extent to which his work derives from that of
Alfarabi. Even though Averroes came to his logic by way of the Farabian treatises, he
went on to distance himself from Alfarabi as his sense of the Aristotelian tradition began
to emerge more clearly. In his early days, Averroes wrote in one of his short works on
Physics that people wanting to understand the discipline should first learn some logic,
preferably from one of Alfarabi’s books [Elamrani-Jamal, 1995, page 51]. But at the end
of his scholarly life, Averroes had reached a different assessment of Alfarabi, an assess¬
ment which we find in his Essays:
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563
One of the worst things a later scholar can do is to deviate from Aristotle’s
teaching and follow a path other than Aristotle’s—this is what happened to
Alfarabi in his logical texts, and to Avicenna in the physical and metaphysical
sciences. ([Averroes, 1983b, page 175.6-8]; cf. [Elamrani-Jamal, 1995, page
52])
Averroes, that is to say, decided that Alfarabi was not Aristotelian enough (at this specific
point, due to Alfarabi’s ampliation of the subject-term and the resulting misformulation
of the dictum de omni ; see above page 549 f.).
As I say, Averroes’ statements about Alfarabi can be misleading. As will emerge in
the next two sub-sections, Averroes followed Farabian inspiration for important elements
that feature in all of his syllogistic systems, and that become characteristic of systems
which may be termed Averroist. We are faced with the irony that Avicenna claimed
Alfarabi as his only worthy predecessor writing in Arabic, and then differed from him in
every major point in the syllogistic, while Averroes upbraided Alfarabi’s logical mistakes,
but developed ideas he found in Alfarabi’s writings into some of his most influential
contributions to logic.
Averroes on absolute and modal propositions In two essays in particular, Averroes
may be seen to be working under Farabian inspiration, and against the Avicennan system.
One of these essays is given by its editor the title, A criticism of Avicenna’s doctrine on
the conversion of premises, the other, On the absolute proposition.
In his essay on the conversion of propositions, Averroes developed his distinction be¬
tween reading a proposition, and more specifically a term, as either perse ( bid-dat ) or per
accidens ( bil-'arad) (a distinction noted and studied in [Lagerlund, 2000, pages 32-35]
and [Knuuttila, 1982, pages 352-353]). This strategy is motivated in the first place by the
conversion of the contingent proposition, which, to preserve the Aristotelian claim, has
to convert as a contingent proposition. The discussion opens by considering the coun¬
terexample all men are contingently writing, which on the face of it should convert to
some who write are necessarily men. Averroes developed his solution, and then went on
to consider the parallel discussion between Avicenna and Alfarabi, which was directed to
the counterexample all animals are contingently sleeping (see above page 550). Alfarabi
tried to save the conversion as a contingent proposition by reading it with a reduplicative
phrase: all animals are contingently sleeping in so far as they are sleeping. Avicenna
rejected this move, arguing that the proposition with the reduplicative phrase is not the
same as the original proposition which was to be converted. Averroes argued that, on the
contrary, the original proposition implicitly contains the reduplicative phrase,
because the animal can only be sleeping in so far as it is sleeping and not in so
far as it is a horse or a donkey or the various other species which sleep. Since
this is the case, the condition is implicit whether it is expressed or not. The
two propositions are one and the same, I mean, that in which the condition
is expressed and that in which it is not. The fact that the condition is part of
the predicate is self-evident, because the animal is not contingently sleeping
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in so far as it is actually sleeping; but rather, if it is [sleeping], then [it is so
due to an aspect had] in potentiality.
If this is the case, then the one actually sleeping in so far as it is actually
sleeping is contingently an animal. But it is accidental for it that if it is an
animal necessarily in so far as it is sleeping potentially, then it is a neces¬
sary* animal. So this premise is necessary per accidens and contingent per
se. So if we say every [creature] sleeping is an animal, and we understand
from it every [creature] potentially sleeping, it is necessary per se\ but if we
understand from it every [creature] actually sleeping, then it is necessary per
accidens, contingent per se. Since every animal is contingently sleeping has
the sense that it is contingent that it is sleeping actually, not potentially, then
were we to understand from it the one sleeping potentially, the premise is
necessary not contingent. Thus one must understand in the conversion of the
contingent the [creature] actually sleeping, and in the conversion of the nec¬
essary the [creature] potentially sleeping. This view is correct, and it contains
the solution to the doubt raised regarding the conversion of the necessary as
a necessary. [... ]
As for the doubt raised relative to the necessary, the solution is known from
what Alfarabi said regarding the possible.
This doubt had to be singled out for treatment due to the prominence of Avi¬
cenna’s questioning of it. [Averroes, 1983b, pages 104.4-105.apu]
I think Alfarabi equivocated in his modal usage between the convertend and the converse,
and I think that Averroes did too in his modified version of the solution (which I have
omitted from the quotation above). Still, the distinction between per se and per accidens
is important in logical systems inspired by Averroes, and this passage serves to show that
Averroes followed Alfarabi in adopting the distinction, the operation of which he then
extended.
Averroes was, in this important respect, Farabian. He was, more importantly, not Avi-
cennan in his modal logic, which he built on top of the assertoric syllogistic. In his Essay
on the absolute proposition, he set down Avicenna’s conditions for reading a necessary
proposition [Averroes, 1983b, pages 120.11-121.5] and how they relate to different def¬
initions of the absolute; “this is all just confusion and disorder” [Averroes, 1983b, page
122.1], Averroes’ own positions on the assertoric as he conceived it over his career are too
complex to be stated compendiously. Two aspects of his position may however be noted.
The first is that Averroes wanted to keep the conversions set down for the assertoric in
the early books of the Prior Analytics, and thought that the distinction between the per se
and the per accidens reading would help him. The second is that he was forced ultimately
to admit that Aristotle’s examples of assertoric propositions could not all be fitted on to
the same set of truth-conditions, and he came at one point to speak of three assertorics,
the mostly-assertoric, the leastly-assertoric and the equally-assertoric [Averroes, 1983b.
pages 117.apu-l 18.2]. Both of these aspects of the Averroist position were considered
subsequently by at least one logician working within his tradition.
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Averroes on the hypothetical syllogistic Like Avicenna, Averroes was concerned to
investigate the interrelation between the categorical and hypothetical syllogistics. His
essay on this. Discourse on the categorical and hypothetical syllogistic, with a criticism
of the conjunctive syllogistic of Avicenna, is deep and complex, and awaits serious study. I
want merely to note some superficial points regarding the lines of argument he developed
to underline another broad aspect of his logic which is Farabian.
Firstly, and unsurprisingly, Averroes’ major goal in the essay is to show that Aristotle
was correct in his views on the hypothetical syllogistic, and that “these syllogisms are not
to be analysed into the figures” ([BadawT, 1948/52, vol. 1, pages 217.u-218.2j; cf. Prior
Analytics 50 6 2—3). Averroes took himself to have proven that the hypothetical is indeed
ineliminable and irreducible to the categorical, and his essay concludes:
So it has become clear that every syllogism and every syllogistic discourse is
either hypothetical or categorical or a compound of the two (and that is called
reduction (half)) according to what Aristotle said in the Prior Analytics. And
that is what we intended to explain. [Averroes, 1983b, page 207.apu-u]
But Averroes was not merely interested in proving Aristotle right. He also wanted to
defend Aristotle against any charges of carelessness in not treating the hypothetical syllo¬
gistic more fully. He did not, according to Averroes, because the hypothetical can prove
no primary Question, and is consequently redundant in scientific writing:
For this reason, Aristotle discarded it and did not set it down in the Prior
Analytics, since his primary intention in the Prior Analytics was to enumerate
the syllogisms essentially useful in demonstration. [Averroes, 1983b, page
197.6-8]
There is much here with which Avicenna would have agreed, even though he belonged to
the tradition which believed that the hypothetical had sufficient importance that Aristotle
had written another treatise devoted to it.
Secondly, Averroes was able to deal with what Alfarabi had written on the hypotheti¬
cal briefly—he thought that Alfarabi was sloppy with his terms, and should have attended
more carefully to an important distinction.
It is clear from what we have said that the kinds of real hypothetical syllo¬
gisms are only syllogisms equivocally. The correctness of what Aristotle said
emerges, that by them no unknown Question is made evident, and that they
are properly part of the Topics. The commentators are agreed on this point.
But their statements become confused when answering why [Aristotle] left
the hypothetical syllogistic out of the Prior Analytics. What [Aristotle] said
regarding them is that they do not produce a primary Question, and they be¬
long properly to the Topics. It appears that this sense relative to the matter
of the real hypothetical did not become clearly distinguished for them; we
find Alfarabi saying in his Posterior Analytics: “As for those demonstrations
composed in the hypothetical, the relation of their parts is the relation of
those composed in the categorical.” But the causes [for production] in the
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hypothetical are the repeated parts of the premises. These [inferences that
Alfarabi is talking about] are not real hypothetical, but merely hypotheti¬
cal by equivocation. Since he did not distinguish this matter with regard to
them, he was therefore not separated from this doubt. [Averroes, 1983b, page
197.9-18]
Lastly, Averroes was not prepared to recognize the Avicennan system of conjunctives
and exceptives (see above page 545 f.). Some of his arguments have to do with epistemic
matters, and are philosophically the most interesting part of the essay, though beyond the
scope of this chapter. This part of the essay concludes:
Most of the well-known book of this man is full of this sort of thing, both
relating to matters logical, and to other matters. Whoever wants to begin in
these arts should avoid his books, for they will mislead rather than guide him.
[Averroes, 1983b, pages 199.U-200.3]
Averroes then moved on from epistemic claims to the division of the syllogistic into con¬
junctives and exceptives, first offering a summary of its propositions [Averroes, 1983b,
pages 200.4-202.8], He tried to show that the conjunctives all collapse into categoricals.
The wonder is that Avicenna posited both these matters together, I mean,
he conceded that every hypothetical premise can be reduced to a categorical
premise (and similarly that every hypothetical Question can be reduced to a
categorical), and then went on to posit that syllogisms composed of hypo¬
thetical are different from syllogisms composed of categoricals. [Averroes,
1983b, page 205.18-21]
This is sufficient for present purposes: Averroes felt able to rectify the Farabian system,
but was convinced that Avicenna’s system puts forward redundant propositions which can
more perspicuously be eliminated.
Averroist logicians I am not sure whether anyone has assembled all the elements of
a system with which Averroes would have been content at one or other stage in his
life. Important elements in four approaches he followed at various times in his career
are presented in [Elamrani-Jamal, 1995], and one systematic overview has been given
in [Manekin, 1993]. We know that structurally similar systems, especially that of Kil-
wardby, came to be important in the middle of the thirteenth century in the Latin West,
although the route by which they got there is not entirely clear [Lagerlund, 2000, pages
32-35]. Debate still goes on about how Averroes’ texts were transmitted [Burnett, 1999];
whether the Jews were the only path for that transmission matters rather less than the fact
that they were one path. Whatever, Averroes’ contemporary Maimonides (d. 1204) was
not part of this process—Efros hesitates between whether Avicenna or Alfarabi was the
greater influence on Maimonides [Maimonides, 1937/38, pages 19-21]. (I doubt whether
the form of Maimonides’ tract is such that we can ever really answer that question.) But
later Jewish scholars such as Gersonides (d. 1344) were certainly reading Averroes, and
adopted many, though not all, of his solutions.
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Levi ben Geshom said: Inasmuch as we saw some things in Aristotle’s Book
of the Syllogism as understood by the philosopher Averroes that appear to us
to be incorrect—namely, in the conversion of modal sentences and the mode
of the conclusion of modal syllogisms, simple and mixed—we have seen fit
to investigate the truth of these matters in this book. [Manekin, 1992, pages
53-54]
Even more important than the lines of transmission is the fact that Averroes was trans¬
missible at all. Because he addressed himself so directly to the Aristotelian corpus, he fell
squarely within the problematic on which the Latin scholars were fixing their attention.
This is why Averroes figured so much more in the West than the Avicennan logicians
ever did. It was not a question of availability of texts, but of common interest. Latin and
Hebrew writers, however, fall outside the confines of this chapter.
By contrast, Averroes rarely appears in later Arabic treatises on logic. One of those rare
appearances is in an epistle on modal propositions which has been edited [El-Ghannouchi,
1971], but never studied beyond that edition, to the best of my knowledge. Ibn-MalTh ar-
Raqqad, about whom we know nothing beyond the name, wrote an epistle, On absolute,
possible and necessary propositions, in which he referred to Avicenna and Averroes and
their solutions to the various problems mentioned above.
This is a strange little text, and I am not sure if it merely relays the stock Baghdad
response to Avicenna’s counterexamples, or something different and more developed.
Whatever the provenance of its doctrine, the epistle is motivated by the fact that “people
have raised doubts against Aristotle regarding the conversion of propositions, especially
Avicenna” [El-Ghannouchi, 1971, page 207.u], doubts which can be laid to rest by induc¬
tively ascertaining the matter in Aristotle’s examples and limiting his claims by these non-
formal criteria. In answering Avicenna’s objections, however, Averroes has been driven
to distinctions which are unAristotelian. On one of Averroes’ distinctions regarding the
absolute, Ibn-MalTh says
All those who sought to solve this problem imposed on it matters which
are not fitting for the doctrine of Aristotle, especially Averroes. He did not
conceive the absolute proposition [properly], and in consequence he made
three kinds of absolute: the most-part, the least-part, and the in-between, as
is the situation with the possible. [El-Ghannouchi, 1971, page 209.20-21]
These distinctions mean that the Averroist absolute “is not the absolute of the Philoso¬
pher” [El-Ghannouchi, 1971, page 209.24]—sufficient grounds to reject the Averroist
absolute. The cult of Aristotle did not die with Averroes, at least not entirely; nor did its
members entirely concur with Averroist doctrines.
5.2 Reseller’s‘Western school’
This and the following section sketch the interests and pedagogical affiliations of the
post-Avicennan logicians in Baghdad and the realms east of Baghdad. As mentioned in
the introduction to this chapter, this is where I think Rescher’s historical model of Arabic
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logic is least helpful and needs to be set aside. I recapitulate the important elements of
that account, because doing so provides the point of departure for this and the next sub¬
section. I offer what seems to me to be the correct version of developments below (see
page 579).
Reseller’s history of later Eastern logic According to Rescher, from the mid-eleventh
century Avicenna’s writings determined the course of philosophical doctrines and discus¬
sions in the realms east of Baghdad. But in the early twelfth century, a major philosopher,
Abu-l-Barakat al-Bagdadl (d. 1165), began systematically to challenge these doctrines.
The spirit in which he did so is characterized as Farabian, which in turn made his logical
writings somewhat resemble the work of the Andalusian logicians. Further, his impact
was such in Baghdad and further east that one may speak of a school of logicians once
again writing in the way of Alfarabi or, at least, challenging Avicenna’s logic in ways
congenial with Alfarabi’s writings. One of the most important scholars to be influenced
by Abu-l-Barakat was FahraddTn ar-RazT (d. 1210), a prolific Western logician; he taught,
among others, Afdaladdln al-KasT (d. c. 1213). So powerful was the influence of the
Western refutation of Avicennan logic that it was not until NasTraddm at-TusT (d. 1274)
that a convincing set of counter-arguments in support of Avicennan logic was produced.
In constructing these counter-arguments, TflsT was working within what Rescher calls the
Eastern tradition, badly debilitated but holding on to solutions of, among others, ‘Umar
ibn-Sahlan as-SawT (d. 1145). Such was TflsT’s achievement that by the early fourteenth
century there were two great schools, the Eastern and the Western. The reconciliation
of these two schools in the fourteenth century by (among others) Qutbaddln ar-RazI at-
TahtanT (d. 1365) was the most important logical event of the period.
There is an abbreviated variant of this history given in another study by Rescher
[Rescher, 1967a], which dates the origin of the Western school to RazT. The abbrevi¬
ated version may be rejected for a subset of the reasons that lead to the rejection of the
longer account in [Rescher, 1964].
Obviously, if there had been a distinct Western school of logicians including RazT and
KasI, this would be an important consideration in setting about the study of their logical
writings. The account of the Western and Eastern schools, however, suffers from some
problems, and considering these problems serves to reveal a more complex reality. Briefly,
the problems are that (1) Abfl-l-Barakat was not simply reviving Farabian logical doctrine,
or even mainly reviving such doctrine; (2) there is no record of a school originating with
Abfl-l-Barakat; but anyway (3) RazT did not follow his doctrine, at least not in the modal
or the hypothetical syllogistic. There is no pedagogical succession from which one may
discern a Western school beginning with Abfl-l-Barakat and being carried on by RazT.
In any event, the Eastern school was not clearly distinct from the Western in terms of
doctrine; (4) RazT was closer, logically speaking, to TflsT than to either Abfl-l-Barakat or
Alfarabi. Further, (5) RazT was actually an important, if not the most important, route
by which TflsT came to receive and understand earlier logical writings—not merely the
writings of Alfarabi, but those of SawT as well. Examining the way that the logicians in
question relate to one another and to earlier scholars reveals a picture more complex than
that of two schools clashing. In the remainder of this sub-section, I examine problems (1)
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and (2) with Rescher’s account noted above, and I examine the further problems in the
following sub-section.
Abu-l-Barakat al-Bagdadl’s logic Abu-l-Barakat al-Bagdad! (d. 1165), who is referred
to by Rescher as Ibn-Malka, was Jewish by birth, from a small town near Mosul. Abu-
l-Barakat moved to Baghdad and, in old age, he converted to Islam. His great work is
The tried and tested book, apparently modeled loosely on Avicenna’s Cure. He interacted
with the Avicennan tradition in complex ways, some of which will appear in what follows.
According to TanakabunT, a nineteenth-century Persian writer, his work had a major im¬
pact on FahraddTn ar-RazI, most especially on RazT’s Eastern investigations ( al-Mabahit
al-masriqiyya), a work which does not touch on the logic. Further, on this account, if
NasTraddTn at-TusI had not countered the writings of RazT, Avicennan philosophy would
have been discarded [Pines, I960-]. Whatever truth there may be in this account for the
history of Islamic philosophy generally, it does not hold for the logic.
And so to aspects of Abfl-l-Barakat’s logic. Abu-l-Barakat mentioned, but was largely
indifferent to, the wasfi readings; most of what he did has to do with modals in the datT
reading. Secondly, though he was working in ways that are influenced by and yet dif¬
ferent from both Alfarabi and Avicenna, Avicenna is incomparably the predominant in¬
fluence. No one after Avicenna could contribute to the Arabic logical tradition without
paying attention to what he had written. But Abu-l-Barakat did more than merely men¬
tion Avicenna to refute him—he accepted a number of arguments and inferences from the
Avicennan account, implicitly rejecting the related Farabian arguments.
One of the best opportunities to examine a move typical of those Abu-l-Barakat made
is in his argument to save the e-conversion of the absolute proposition against Avicenna’s
counterexample (see above page 548 ff.). Abu-l-Barakat warned people against being like
one
according to whose view, imprecise as it is, the e-proposition does not convert
as an e-proposition (as Aristotle had said). He gives an example for that view:
laughter may be negated of every person actually at a certain time, so that is
an absolute negation. Yet it does not convert, that is, its converse is not
true (that no one laughing is a man, for rather, everyone laughing is a man).
But he has not considered his words ‘at a certain time’ and ‘actually’. The
absolute is absolved ( mutlaq ) of these and other matters; no given time is
mentioned in it, nor any condition. Rather the predicate and the subject are
mentioned, and the quantifier in an affirmative, and the particle of negation
in a negative, without anything further. If [the proposition] is said like that,
then the example offered is not credible, since no one who conceives things
accurately on hearing it would accept no man is laughing as an absolute
statement because [each man] is not laughing at some times, while he would
accept that every man is laughing because [each man] is laughing at some
times. So the form of the words in affirmation does not convey perpetuity,
yet in negation the form does convey perpetuity, such that the negation has to
be a negation in accordance with that.
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Reflect on these words, and how they fall in with comprehensibility and
conceivability—dispense with all they go on about, and ascertain the cor¬
rectness of Aristotle’s doctrine in his extremely clear words that do without
the subtleties used above. [Bagdad!, 1357 A.H., pages 120.20-121.8]
Abu-l-Barakat accepted the Avicennan truth-conditions for the a-proposition in the ab¬
solute, and merely argued against rejecting the ‘conventional’ truth-conditions for the
e-proposition (that is to say, his a- and i-propositions are like proposition 10 in appendix
two, and his e- and o-propositions are like proposition 5). Abu-l-Barakat preserved the
immediate inferences of the Aristotelian assertoric, but he did it while accepting Avicen¬
nan truth-conditions for the absolute a-proposition.
So much for the first move in the Avicennan modification of the assertoric syllogistic.
Abu-l-Barakat went on to give the accounts not only of conversion, but also of assertoric
contradiction, to be found in the first seven books of the Prior Analytics. He then moved
on to the modals. Unlike Alfarabi, and after him Averroes, however, Abu-l-Barakat did
not use reduplicative propositions to get the conversion of the two-sided possible as a
two-sided possible. In fact, he came to the same conversions for his modals that Avicenna
proposed, rejecting those defended by Alfarabi and Averroes [Bagdad!, 1357 A.H., pages
121-122]. Abu-l-Barakat later in his treatise gives the syllogistic moods with both the
assertoric second figure, and the fourth figure [Bagdad!, 1357 A.H., page 125 f.; page
148 ff.].
Abu-l-Barakat accepted the Avicennan division of the syllogistic into conjunctive and
exceptive. He also quantified and negated his conditional propositions like Avicennan
conditionals. He was largely indifferent, however, to the hypothetical syllogistic, and
belonged to the tradition sceptical of positing a lost ‘Aristotelian’ treatment of the hy¬
pothetical syllogistic, a tradition represented more than two centuries before by Alfarabi
(see above page 543).
Regarding syllogisms which are from hypothetical propositions, Aristotle
only made mention of the exceptive in his book. What touches on conjunctive
[hypothetical] syllogisms, both pure, and mixed with categoricals, is clear
from what he says, and the sound mind will recognize them from what has
been said. He omitted mentioning them in his book either due to how little
benefit they are in the sciences, and he disliked the thought of dwelling on
them; or because he relied on the fact that minds which have come to know
the categoricals may conclude from them to [the hypotheticals], so that you
will recognize them from what you have come to know in the categoricals;
or [he omitted mention of them] for both [reasons], A certain later scholar
said that Aristotle had written a special book on them which had not been
translated into Arabic; this is baseless conjecture. Had he wanted to mention
them, why did he move them from here, their proper place? Anyway, there is
not enough concerning them that would merit a separate book with separate
principles and conclusions. [Bagdad!, 1357 A.H., page 155.11-18]
Further, Abu-l-Barakat did not analyse the proofs by reduction using the distinction
[Bagdad!, 1357 A.H., page 188.11-12],
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We can hardly claim that Abu-l-Barakat represents a modification of the Avicennan
system in the spirit of Alfarabi. He did not adopt any of the Farabian solutions attacked by
Avicenna in The Cure (noted above, page 547 ff.), and he accepted the modal conversions
that Avicenna specified. He also accepted the Avicennan division of the syllogistic into
conjunctive and exceptive, though he did not emphasize or use it as much as Avicenna
did. It is true that he was not like other Avicennan logicians, described in the next section,
who were particularly interested in the extensions of the wasfi readings. Nor was he a
precursor to what Averroes did, and although an Aristotelian, his Aristotelianism is much
more textually attenuated than that of Averroes.
‘AbdallatTf al-Bagdadl That said, there is at least one scholar in Baghdad, somewhat
later than Abu-I-Barakat, who did direct his philosophical project towards a recovery of
Aristotle by way of Alfarabi. This philosopher was the rather idiosyncratic ‘AbdallatTf
al-BagdadT (d. 1231), who, having studied Avicennan philosophy, went from Baghdad
westwards, travelling widely. On his travels, he met scholars who convinced him that the
philosophy he had studied was not as good as Alfarabi’s. He came to write more along
Farabian than Avicennan lines, and apparently composed a number of commentaries on
Alfarabi’s works [Gutas, 1993, page 50], This is very interesting, although it must be said
that we do not to this day have in published form a logical treatise by ‘AbdallatTf. What we
do have is a manuscript in Brusa which may well be vital, not merely for understanding
‘AbdallatTf, but also for reconstructing Alfarabi’s lost Long commentary on the Prior
Analytics. The relevant part is paraphrased by Stern as follows:
Some particular points in Aristotle’s logic have been criticised, but it turned
out he was right and his critics did not understand his meaning; this has been
explained by al-FarabT in his great commentary on the Prior Analytics. It
is altogether a great mistake to think that the modern works are clearer in
exposition or style than those of the ancients. [Stern, 1962, page 63]
Until we reassemble the logical writings of ‘AbdallatTf, we can do little more than note
that at least one scholar worked in a Farabian rather than an Avicennan line.
5.3 The Avicennan tradition
Neither Abu-l-Barakat nor ‘AbdallatTf represents a larger school which had returned to
Farabian doctrine, at least in the syllogistic. Nor was FahraddTn ar-RazT the representative
of a school at war with the school to which the great NasTraddTn at-TusT belonged. In
dealing with this second set of objections to Rescher’s claim for a Western school (that
is, objections 3, 4 and 5 at page 568 above), I propose to show two things: in terms of
the later reception of Avicennan logic, RazT was rather more one of TusT’s sources than a
target for criticism, and, secondly, in terms of substance, RazT and TusT were interested in
broadly the same questions and came to roughly the same answers. In short, I approach the
material here firstly source-critically, and then in terms of its substantive logical doctrine.
572
Tony Street
RazI, TusT and the logical tradition In coming to terms with the scholarly relation
between RazT and TusT, one has first to negotiate a tendency in studies on the history
of Islamic philosophy to overstress the differences between the two men. This arises
in large part because of the excoriating attacks made by TusT in commenting on RazT’s
Validated philosophy of the ancients and the moderns (Muhassal afkar al-muta' ahhirm
wal-mutaqaddimln), a text which does not deal with the logic. But TusT had at least a
grudging admiration for RazT’s commentary on Pointers. This was so in spite of the fact
that it is often said in histories of Islamic philosophy that TusT thought RazT’s commentary
was a “diatribe not a commentary” [Fakhry, 1983 2 , page 320, to cite one of many possible
examples]. What TusT wrote is this:
Among those who have already commented on this book is the eminent and
erudite FahraddTn, prince of the controversialists, Muhammad ibn-‘Umar
ibn-al-Husayn al-Hatib ar-RazT. He made an effort to explain as clearly as
possible everything in it that was hidden, and strove to express in the best
way that which was obscure; he followed in hot pursuit of what was meant,
and, in searching out what was deposited therein, he reached the furthest path
of penetration. He was excessive, however, in responding to Avicenna in the
course of his essay, and in refuting his principles overstepped the bounds of
justice. By these efforts he only detracted from his own work, and because
of them a certain wit has called his commentary a diatribe. [TusT, 1971, page
112 . 1 - 6 ]
TusT thought RazT’s commentary was overly oppositional in expositing Avicenna’s sys¬
tem, but when TusT was asked to write a commentary on Pointers himself, he said of its
notoriously laconic doctrinal payload: “I gained it from the first commentary, mentioned
above, and from other famous books...” [TusT, 1971, page 112.18-19].
I have dwelt on this because I think it is why Rescher came to decide that RazT and
TusT were at loggerheads in the logic. They were not, even though they may have been
in metaphysics. More important than any explicit appraisal of RazT by TusT, however, is
how they both approached the logical tradition and its problems. As it turns out, TusT
named the scholars he was drawing on, and how he differed from them. The logicians
to whom TusT referred may be divided into three groups, groups which are mentioned
to serve different functions in the course of TusT’s exposition. The first group of logi¬
cians he mentioned consists of Greeks from classical times and late antiquity: Aristotle,
Theophrastus, Eudemus, Alexander, Themistius, Porphyry. The second group, or really
pair, of logicians mentioned is made up of Avicenna and Alfarabi (TusT rightly took ‘the
eminent later scholar’ of The Cure to be Alfarabi). There are, lastly, four scholars whose
names are invoked, who either died somewhat before TusT was writing, or were his con¬
temporaries. The oldest of these scholars is ‘Umar ibn-Sahlan as-SawT (d. 1145) [Brock-
elmann, 1936-1949, Supplementary volume I, page 830], who wrote Insights into logic
for Naslraddln [SawT, 1898]. The next oldest source quoted by TusT is RazT. The third
logician is AfdaladdTn KasT—death-dates for KasT vary widely, from 1213 or 1214 to the
early fourteenth century, but the earlier date seems preferable [Chittick, 1982-]. Rescher
has KasT as RazT’s student, though there is little evidence to support this [Rescher, 1964,
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573
page 68]. KasT’s second longest work is an Arabic treatise on logic called the Clear Path
[KasT, ]. The last of the logicians mentioned is AtTraddln al-AbharT (d. 1264); Abharl
studied under KamaladdTn ibn-Yunus, who was probably also TusT’s teacher. Abharl
wrote a number of logical works, among them the famous Introduction to logic [Calver-
ley, 1933]. Sadly, I have not seen any of AbharT’s longer works on logic—I think none
has been printed.
I list these sources because it allows us to examine which sources TusT shared with
RazT, and whether he read them in the same way that RazT did. RazT did not mention
the Greek logicians (at least on my reading), but he did mention both Alfarabi and SawT.
Before I turn to how RazT read these logicians, and influenced TusT’s reading of them, I
should say that although RazT did not mention the Greeks whom TusT mentioned, he did
deal with all the logical doctrines that TusT addressed in referring to the Greeks. (I return
to this point; see below page 577 f.)
RazT referred to Alfarabi by name to bring out his doctrine of the ampliation of the
subject-term. I don’t think he made such a reference in Gist or his longer commentary
on Pointers , but in his Summary of philosophy and logic, he wrote that “Alfarabi claimed
that in all Js one should not consider the occurrence of actual Js, but rather what may
be describable as J” [RazT, a, folio 23a. 13]. In the longer commentary on Pointers itself
he gave the Alexandrian proof for e-conversion without, however, ascribing it to anyone
[RazT, b, folio 4lb. 12 et seq.]. TusT gave accounts of precisely these Farabian doctrines
(the ampliation of the subject-term [TusT, 1971, page 282.1-4]; e-conversion [TusT, 1971,
page 325.8-12]). Both men, that is to say, were reading exactly the same things out
of Alfarabi, or perhaps relaying the same things—I wonder if either had actually read
Alfarabi.
Again, TusT read SawT the same way RazT did. At every point that TusT consulted SawT
in his commentary, RazT had preceded him. In treating the conversion of the wasfl non-
perpetual, TusT wrote almost verbatim [TusT, 1971, page 328.5 et seq.] what we find in
RazT [RazT, b, folio 41b.pu et seq.]; so too for the first-figure syllogisms with mixed wasfl
and datl premises (compare [TusT, 1971, page 400.12 et seq.] and [RazT, b, folio 53b.pu et
seq.]), and for the second-figure syllogisms with the same mix (compare ITusT, 1971, page
416.19 et seq.] and [RazT, b, folio 57b. 18 et seq.]). In fact, I doubt that TusT had actually
read SawT for himself, because he credited RazT with coining the term ‘conventional’
(‘urfiyya), even though we find it in SawT [SawT, 1898, page 73.5] (although there of
course it may be being used pre-technically).
None of this is surprising; it is exactly what TusT announced he would do at the be¬
ginning of his commentary. To picture TusT and SawT in an Eastern school, doing things
logically different from RazT—this just misrepresents what was going on. Further, RazT
and TusT were both closer to each other than either of them was to Avicenna. Clinging to
the idea of an Eastern and a Western school makes it difficult to understand why schol¬
ars who appear as either Eastern or Western are doing such similar things (and things so
different from what Abu-l-Barakat was doing), and why ‘Easterners’ draw so freely on
‘Westerners’ and vice versa. Rescher’s account of the Western school [Rescher, 1964,
page 57] needs to be rejected.
574
Tony Street
Along with the rejection of the ‘clash of the schools’, we need further to reject the
reconciliation of the schools in the Arbitration between the two commentaries [TahtanT,
1375 AH solar] by Qutbaddin ar-RazT at-TahtanT (d. 1365). (My comments here do not
in any way go to what TahtanT may or may not have been doing in the physics and meta¬
physics.) There are about forty references in TusT’s commentary on the logic of Pointers
to RazT, and they clearly did differ on many points. But following TahtanT’s Arbitration
as it goes through the logic, one cannot fail to be struck by how often TahtanT had nothing
to say on the points of difference and, on those few occasions he did say something, how
rarely it consituted a synthesis or reconciliation of two seemingly incompatible views. As
it happens, attitudes of some logicians in Iran and further east differed from the attitudes
of others, in a way I will examine below. There were not, however, two schools differing
fundamentally on matters of substantive logical doctrine.
No logician after Avicenna defended the Avicennan syllogistic pure and simple.
Changes were put forward for (among other things) the dciti propositions, the wasfi propo¬
sitions, the temporals ( waqtiyyat ), and the analysis of the proof by reduction. I note some
aspects of each in turn, but I stress that these notes fail badly in conveying the depth and
range of logical analysis of each aspect, because they are limited to the discussions which
issued in the doctrine of the madrasa texts. It is this development of logic which was said
by Ibn-Haldun to be penetrating, constituting a discipline in its own right (see below page
580); the same development was compared by Ibn-Taymiyya to carrion (see above page
560).
Datl propositions It was noted (see page 552 above) that Avicenna’s syllogistic with
dati premises includes some puzzling inferences or, more strictly, sets of puzzling infer¬
ences. Take, for example, the claim that absolute a- and i-propositions (see proposition
10 in appendix 2) convert as absolute i-propositions, that one-sided possible a- and i-
propositions convert as one-sided possible i-propositions, and yet that syllogisms with
possible minors and absolute majors produce. In fact, syllogisms with possible minors
are central to the development of Avicenna’s system. There may well be a way to show
the compatibility of all the inferences Avicenna proves, but I cannot see what it is. More
importantly, nor could the thirteenth-century logicians. Early on, efforts were made by
FahraddTn ar-RazT to save most of Avicenna’s modal syllogisms, seemingly by taking
the subject-term to ampliate to the possible. This meant having the absolute a- and i-
propositions convert as possibles (see e.g. [RazT, 1355 2 A. H., page 24.16-20]).
I am not sure if the inferences in RazT’s alternative system actually square any better
than Avicenna’s, but in any event, no one seems to have adopted his approach. By the
time TusT’s student NagmaddTn al-KatibT (d. 1276) was writing, it is clear that scholars
had come to agree that syllogisms with possible minors do not produce, at least not if the
propositions are read according to the ordinary way of taking the subject-term. KatibT
put forward the following distinction: In a world where it so happens that there is no
geometrical figure apart from triangles, all figures are triangles is true. If. however, we
are concerned not with how things happen to be, but with how meanings relate, all figures
are triangles is untrue, even in that world where it so happens that all figures are triangles.
The subject-term was to be read in the first way.
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575
TusT and KatibT differed from RazT over how to take the subject-term, and the KatibT
view won out, at least in the later tradition that has so far been examined in modern
studies. In Muhammad ibn-Faydallah as-Sirwanl (fl. 15th century?), for example, we find
Avicenna characterised as taking the subject-term as referring to that which is “actual,
that is, at a given time, whether it be at the time of the judgement, or in the past, or in the
future” [SirwanT, , folio 96a.6-10], a characterisation which would be less problematic
if it were not directly linked to the untrue claim that Avicenna did not allow possible
affirmative propositions to convert. Strange to say, RazT was more Avicennan than KatibT
on this point. It is a major point, affecting the fabric of the entire dati syllogistic; KatibT’s
decision means that all first-figure syllogisms with possible minors fail to produce, and
all the syllogisms which depend on these mixes for their proofs also fail. And RazT was
not just defending syllogisms with possible minors in an act of commentatorial charity;
we find him committed to them not only in his commentaries on Avicenna [RazT, 1355 2
A. H., pages 33-34] but also when he is speaking in his own voice, in The summary of
philosophy and logic [RazT, a, folio 46b.9].
We should note one other point of comparison between the thirteenth-century logicians
and Avicenna in their presentation of the syllogistic with dati modal premises, which has
to do with their respective concern for Aristotelicity. Avicenna mentioned the perpetual
(proposition 5 in appendix two) in his development of the syllogistic only when giving
the contradictory of the absolute, but at no other place. I can only speculate about the rea¬
son for this, but I think it is because Avicenna’s presentation is developed in conversation
with Aristotle’s account, and thus he can find no place for a perpetual, because there is no
perpetual in the Prior Analytics. Whether or not I am right about this, it is none the less
true that the short thirteenth-century accounts have the perpetual as a proposition fully
investigated throughout the presentation. Further, some of the converses of dati propo¬
sitions are no longer dati propositions (these points should become clear by comparing
appendices one and two). I think all of this is symptomatic of increasing indifference to
the Aristotelian account (see further page 577 below).
Wasft propositions All the logicians I have read were agreed that there were seri¬
ous problems with Avicenna’s account of propositions in the descriptional reading (the
wasfiyyat). Avicenna claimed, among other things, that the contradictory of all Js are
always Bs while Js is some Js are always not Bs while Js (it is not clear to me that this is
what Avicenna actually claimed, but anyway, this is what he was taken to have claimed).
Among the limited sources I have examined, this concern began with SawT [SawT, 1898,
page 70.10 et seq.], and his approach to the problem is reflected in RazT’s Gist [RazT,
1355 2 A. H., page 22.1-4]. It is interesting to follow the concerns about Avicenna’s
claims that a perpetual in the wasft reading is contradicted by another perpetual in the
wasfi reading as they gradually gather clarity and, ultimately, technical terms to describe
the concerns. By the time of TusT, it was taken as settled that a solution had been reached,
and that the descriptional perpetual (a wasfi reading, proposition 6 in appendix two), must
be contradicted by a wasfi absolute, called by TusT mutlaqa wasfiyya [TusT. 1971, page
313.8] (proposition 7 in appendix two). KatibT called it the hiniyya [KatibT, 1854, page
16.8] , though it is not given in his treatise as one of the propositions in the preliminary list-
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ing. Hiniyya is adopted as the usual term for the proposition in the subsequent literature
[SirwanT,, folio 86a. 12], and it joins the other propositions in the preliminary listing. We
can be sure that AbharT was also working on questions to do with the wasfi propositions,
because TflsT mentioned in passing that he took a conventional existential o-proposition to
convert (this is not a point AbharT makes in his Introduction, and I cannot check it). Avi¬
cenna’s claims regarding the wasfi readings have been the subject of a recent study which
proceeds by adopting an understanding of the proposition proposed by TusT [Thom, ].
In the case of the dati readings and the perpetual, the thirteenth-century logicians were
indifferent to the fact that Aristotle had not used a perpetual in his account in the Prior
Analytics. In treating the wasfi readings, they betray no feeling of pressure to find a propo¬
sition which conforms to the immediate inferences required of the Aristotelian assertoric.
Nor are they primarily interested in syllogisms with purely wasfi premises, which I think
was Avicenna’s primary concern, but with the ways mixes of wasfi and dati readings
produce. This takes up a later and less central concern of Avicenna.
Temporals One of the extrinsic conditions on a proposition which occurred in Avi¬
cenna’s list was the as-of-now. Avicenna used it once in the course of his exposition to
save the Aristotelian account of contradiction. TusT’s student KatibT introduced it into the
propositions “into which it is usual to inquire,” using it to produce a modalized proposi¬
tion. Much later, the temporals were exhaustively analysed by SirwanT (see appendix two,
propositions 3, 4, 8, 9, 12, 14, 17, 18, 22 and 23). As with the wasfi readings, however,
the thirteenth-century logicians were not concerned to use the as-of-now to preserve the
Aristotelian account, and changed its truth-conditions so it no longer squared even with
the way Avicenna used it.
Proofs by reduction Another point that needs to be considered is how the Avicennan
logicians dealt with the proof by reduction. (I will not try to describe their extensive exam¬
ination of the hypothetical syllogistic.) SawT took it to be a combination of a conjunctive
and an exceptive [SawT, 1898, page 104.pu-u], just as Avicenna had. RazT followed SawT
and Avicenna on this point [RazT, 1355 2 A. H., page 43.17-18]. But KasT, on the re¬
lation between the categorical and the hypothetical syllogistics, differed from Avicenna
and offered an alternative analysis which is treated sympathetically and, I think, actually
adopted by TusT and subsequent logicians whom he influenced.
KasT’s argument seems to me to amount to no more than an argument by assertion,
though perhaps more sensitive and acute study of the problem will turn up a different
conclusion. What KasT concluded is:
The deduction involving a contradiction is an exceptive syllogism whose mi¬
nor premise is hypothetical with a compound antecedent, and whose major
premise is categorical, being the contradictory of the consequent. So it pro¬
duces the contradictory of the first proposition of the two parts of the an¬
tecedent of the minor. This is its form:
If Zayd is writing, and everyone who is writing moves his fingers,
then Zayd is moving his fingers;
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But Zayd is not moving his fingers;
Therefore Zayd is not writing. [KasT,, folio 72a. 1-5]
KasT felt compelled to put forward his alternative view of reduction because of arguments
which had become common.
The reason for this disquisition is to alert people to the truth of the matter
concerning how a deduction involving a contradiction is composed. This is
because what is to be understood from a certain verifying scholar is other
than what we have mentioned. He said rather that this is not the case, but that
the deduction involving a contradiction is composed of two syllogisms, one
conjunctive and the other exceptive; just as when it is said that the hypothet¬
ical proposition is composed of two categorical propositions, from which it
is to be understood that the hypothetical proposition is something other than
these two. The deduction involving a contradiction is not like that—it is just
an exceptive; it is an exceptive syllogism whose minor premise is composed
of two categorical propositions sharing a term, from the granting of which
there follows as a consequent of the first proposition with its two parts, and
the major premise, a categorical proposition which contradicts the antecedent
of the minor. This has been determined before, and illustrated. [KasT,, folio
71 b.u—72b. 13]
It is worth noting the form of KasT’s claim. He did not derive his view of the matter from
the Averroist critique, or from its antecedents.
The cult of Avicenna TusT seems to have adopted KasT’s analysis of the reduction. It
is instructive to note what he took to be Avicenna’s reasons for taking the view he did on
the hypothetical syllogistic.
Aristotle placed the deduction involving a contradiction among the hypothet¬
ical syllogisms, yet in his writings there were only exceptive hypotheticals;
consequently most logicians simply called [the exceptive] a hypothetical syl¬
logism.
However, Avicenna thought that the conjunctive hypotheticals had been treated
in a separate text which was not translated into Arabic. [This was] simply an
assumption which Avicenna was compelled to hold due to his good opinion
of Aristotle. When the later logicians wanted to analyse this syllogism, and
reduce it to the above-mentioned syllogisms, that analysis was difficult for
them to accept, and they differed completely from Avicenna. [TusT, 1971,
page 453]
This exemplifies the way that TusT approached the Avicennan system, and how his
approach differed from RazT’s. The difference is even starker when TusT mentioned Greek
philosophers at another important point in his commentary, as will emerge.
TusT was among the many who accepted the modifications to Avicenna’s doctrines
regarding the wasfi propositions. Unlike other scholars, however. TusT did so while at
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the same time presenting Avicenna as responding to an ancient peripatetic debate re¬
garding what place the absolute proposition has in scientific discourse. Avicenna’s kinds
of absolute proposition (corresponding roughly to propositions 6, 10, 20 and 24 in ap¬
pendix 2), TusT claimed, find places for the various doctrines of Aristotle, Themistius and
Theophrastus, and Alexander [TusT, 1971, pages 268.pu-269.20]. The problem is that
the datl reading of the absolute does not find contradictories or convert as an Aristotelian
assertoric, so Avicenna stipulated a reading of the absolute that he thought did. TusT
wrote:
What spurred him to this was that in the assertoric syllogistic Aristotle and
others sometimes used contradictions of absolute propositions assuming them
to be absolute; and that was why so many decided that absolutes did contra¬
dict absolutes. When Avicenna had shown this to be wrong, he wanted to
give a way of construing those [examples from Aristotle], [TusT, 1971, page
312.5-7]
The modified, convertible absolute proposition presented by Avicenna is also able, ac¬
cording to TusT, to accommodate the opposing interpretations of the absolute proposition
put forward by Theophrastus and Alexander.
We have mentioned that the validating scholars of this art had two views in
explaining the absolute. The first of them is that it includes the necessary, as
Themistius held, and that is the general. The second is that it does not include
the necessary, as Alexander held, and that is the special. Avicenna wanted
to show that each of the two views could be specified in the way which he
puts forward here, so that it is compatible with contradiction in the absolute
according to both of the two views.
His explanation is that the conventional can be taken to cover the necessary,
and be general; or it can be taken not to cover the necessary, and be spe¬
cial. The conventional general absolute agrees with the first view; and the
special... agrees with Alexander. [TusT, 1971, pages 313.16-314.6]
On the fact that Avicenna took such pride in the fact that his two readings of the proposi¬
tions allowed him to square a set of examples where other logicians had failed (see above
page 551), TusT wrote:
He means the majority of logicians were not able to escape the consequences
of their doctrine, that is, that absolutes contradict absolutes. This is because
they were not able to construe the absolute mentioned in the First Teaching
in all places according to their doctrine. Among relevant examples in the
First Teaching are the absolutes All who wake sleep and All who sleep wake,
and others like them which cannot be construed as conventional. Similarly in
usage, since the First Teaching used the absolute where it is not possible to
use the conventional. [TusT, 1971, pages 314.12-315.2]
There are no references to Greek scholars in the corresponding passages in RazT's texts
(at least on my reading). Both scholars were convinced that Avicenna’s system needed
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repair by either extension or restriction, and both proceeded to implement such changes,
sometimes TusT more than RazT (especially for example in the datT propositions). But RazT
made his changes with little outward show of respect; at one point in Gist, for example,
we find him saying:
Once you have come to understand what I have mentioned here, you will
realise that this commentary, in spite of its brevity, is more explanatory and
rigorous than what is in Avicenna’s book, in spite of its length. [RazT, 1355 2
A. H., page 22.14]
TusT, by contrast, nearly always found a way to justify the position Avicenna had adopted.
That is the role the Greek authors play in his commentary—they indicate where Avi¬
cenna’s doctrine is open for contestation and reinterpretation, while saving Avicenna from
any charge of logical error.
Post-Avicennan logic in Baghdad and further east At this point, I offer a short and
tentative outline of the history of post-Avicennan logic in Baghdad and further east.
In Baghdad, attention continued to be paid to the ways Aristotle had organised his
logic and proved his inferences, though by 1150 at the latest it was clear that Avicennan
counterexamples had problematized the Aristotelian account. We can assume that Al-
farabi’s texts were being read in Baghdad, though the fact that someone like ‘AbdallatTf
al-BagdadT had to learn his Farabian logic outside of Baghdad shows that the tradition of
reading these texts was growing ever weaker. References to Alfarabi among these logi¬
cians are so stereotypical that we must wonder if his texts were available at all. By the
time of Barhebraeus (d. 1286), even the Syriac Christians had become Avicennan.
Further east, by the first half of the twelfth century, an Avicennan tradition was well ad¬
vanced in the process of modifying Avicenna’s system. The Avicennan tradition may be
called ‘Avicennan’, as has been mentioned, because it accepted the division of the syllo¬
gistic into the conjunctive and exceptive, it accepted the conditions proposed by Avicenna
as the relevant ones within which to investigate modalities and temporalities, and it ac¬
cepted the stipulation of truth-conditions for the absolute such that it was contradicted by
a perpetual. This tradition proceeded with little or no reference to the Aristotelian corpus,
at least in its early days, producing a modified system by the end of the thirteenth century
which by and large continued to be accepted down till well into last century. Among some
of its later adherents, the tradition started to refer once again to the writings of Aristotle;
these references are associated with a cult of Avicenna which used the Greek references as
a way to excuse some of Avicenna’s less easily defended moves. But scholarly courtesy
to the great Avicenna in no way prevented changes being made to his logical system.
5.4 The Handbooks of the Madras a
At roughly the same time that a consensus was emerging that the major questions in Avi¬
cenna’s formal syllogistic had been settled, the madrasa was reaching a period of institu¬
tional stability and influence. GazalT’s opinion that logic was useful for Muslim scholars
was an important factor in having the subject included in the syllabus of the madrasa. The
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doctrine of the madrasa handbooks was determined by the Avicennan tradition described
in the last section. We can be fairly sure that this was the dominant tradition by the mid¬
thirteenth century not only from the fact that even the Syriac Christians used Avicennan
logic, but also from the account of the great polymath, Ibn-Haldun, who wrote about some
of the important handbooks of the time.
The emergence of the handbooks Ibn-Haldun received his training in logic from these
handbooks, and in his Prolegomena, he described both the changes in emphasis over
the years within the discipline of logic, and which books were used in teaching. After
describing the composition of the Organon and the logical issues each part of it addresses,
the Prolegomena continues:
Its sections came to be nine; and all were translated in the Islamic commu¬
nity, and the philosophers dealt with [these books] by commentary and expo¬
sition. Alfarabi did [this], and Avicenna, and Averroes among the Andalusian
philosophers—Avicenna wrote The Cure, in which he took in all seven philo¬
sophical disciplines. Then the later scholars came and changed the technical
terms of logic; and they appended to the investigation of the five universal
its fruit, which is to say the discussion of definitions and descriptions which
they moved from the Posterior Analytics', and they dropped the Categories
because a logician is only accidentally and not essentially interested in that
book; and they appended to On Interpretation the treatment of conversion
(even if it had been in the Topics in the texts of the ancients, it is none the
less in some respects among the things which follow on from the treatment
of propositions). Moreover, they treated the syllogistic with respect to its
productivity generally, not with respect to its matter. They dropped the inves¬
tigation of [the syllogistic] with respect to matter, which is to say, these five
books: Posterior Analytics, Topics, Rhetoric, Poetics, and Sophistical Falla¬
cies (though sometimes some of them give a brief outline of them). They
have ignored [these five books] as though they had never been, even though
they are important and relied upon in the discipline. Moreover, that part of
[the discipline] they have set down they have treated in a penetrating way;
they look into it in so far as it is a discipline in its own right, not in so far
as it is an instrument for the sciences. Treatment of [the subject as newly
conceived] has become lengthy and wide-ranging—the first to do that was
Fahraddln ar-RazT and, after him, HunagT (on whose books Eastern scholars
rely even now). On this art, HunagT has written The Disclosure of Secrets,
which is long, and an abridgement. The Short Epitome, which is good for
teaching, and another abridgement. The Digest, which in four folios takes up
the cruces and principles of the discipline—students use it frequently to this
day and benefit from it.
The books and ways of the ancients have been abandoned, as though they had
never been... ([Ibn-Haldun, 1858, pages 112.8-113.12]; cf. [Ibn-Haldun,
1958, pages 142-143])
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The formal syllogistic, Ibn-Taymiyya’s camel carrion, survived the ending of interest in
the ‘material’ disciplines. Ibn-Haldun dated this change from RazT’s work, though it is
apparent already in SawT’s Insights and even in Avicenna’s Pointers and Reminders.
The other scholar in Ibn-Haldun’s account is Muhammad ibn-Namwar al-Hunagl
(d. 1249), who had been a judge in Cairo. I think the point Ibn-Haldun was trying to
make in mentioning his treatises is that logicians writing in the West were doing logic
so well that even logicians in the East had to take notice. An examination of his Digest
reveals just what Ibn-Haldun meant when he talked about the change in focus and depth
of treatment. Of fourteen pages, the first three are given over to utterances, universal,
definitions and descriptions; all the rest are given over to the formal study of propositions
and the syllogistic. The presentation of the different kinds of propositions begins on 76a,
and proceeds by laying out the modalities and temporalities (necessity and perpetuity and
their duals), goes on to the readings {datX and wasfX), and then the temporal constants (the
temporal and the spread). Mention is made of the different doctrines on the absolute, and
of the different doctrines on how the subject-term can be taken. What we find, in short,
is precisely the same sort of approach to the propositions that we find in RazT and TusI—
though it is important to stress that the doctrine presented is not uniquely that of RazT, but
rather reflective of the main options in his and TusT's tradition. Again, in the division of
the syllogisms, HunagT adopted the Avicennan distinction between conjunctive and ex¬
ceptive, though it is impossible to gauge from his reference to reduction [HunagT,, folio
77a. 11-13] how he analysed it. Proportionally, an extraordinary amount of attention is
devoted to the conditionals [HunagT,, folio 78a-80a].
Two standard texts It was two contemporaries of HunagT, however, who wrote the
treatises most widely used as introductions to the subject over the centuries: AbharT (see
[Gutas, 1993, page 63 note 161]) and KatibT (see [KatibT, 1854]). In Calverly’s transla¬
tion, AbharT’s Introduction comes to eight pages [Calverley, 1933]. In much the same
way as Avicenna’s introductory Philosophy for ‘Ala’addawla [Avicenna, 1971], it is is
very elementary and general. Aside from the fact that it divides the syllogistic into the
conjunctive and exceptive, it could come from just about any tradition derived from Peri¬
patetic logic. But it presents its logic in the order and with the terms that make it the
perfect preliminary to a more difficult text.
For most students through the centuries that text was KatibT’s Logic for Samsaddxn
[KatibT, 1854]. In Sprenger’s edition it comes to twenty-nine pages. It has been trans¬
lated, mostly by Sprenger and Kaye, and by Rescher [Rescher, 1967b]. The whole text is
probably ready for a new translation, complete with annotations and the semantics due to
Rescher and vander Nat [Rescher and vander Nat, 1974] (given in appendix two below).
I offer a rapid sketch of the book.
KatibT did not compare logic to grammar. But he wrote that to acquire the demon¬
strative sciences and come in contact with the angelic intelligences, the loftiest pusuit for
man, one cannot do without logic [KatibT, 1854, page 1.5-7], KatibT went on by intro¬
ducing the terms tasawwur and tasdlq in the first section of his treatise [KatibT, 1854,
page 2.5-7], naming the subject-matter of logic as “the objects of cognition, both concep¬
tual and prone to assent” (al-ma‘lumatu t-tasawwuriyya wat-tasdXqiyya) [KatibT, 1854,
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page 2.18]. The theory of definition deals with conception, while the theory of proof
deals with assent [KatibI, 1854, page 2.u]—this is straightforwardly Avicennan. There
is no reference to the reduction of legal arguments to the syllogism, and the context the¬
ory is confined to the section on syllogistic matter (fl mawaddi l-qiyasdt). That said, the
equation of merely presumed matter ( maznun ) with rhetoric would have signalled clearly
where juristic arguments were thought to fit into the theory.
Like AbharT, KatibI accepted the division of the syllogistic into conjunctive and excep¬
tive, though the text does not devote much attention to conjunctive hypotheticals, nor to
the Avicennan analysis of the deduction involving a contradiction. The text then presents
a version of the assertoric syllogistic that squares with Aristotle’s account, but goes on to
say that this will not work for the absolute proposition. KatibI’s subsequent account of the
syllogistic with the various modalities and temporalities in both the dati and wasft read¬
ings is a modification and extension of Avicennan ideas. Overall, his logic is Avicennan
only in the attenuated sense that TusT’s logic is.
KatibI and his teacher TusI also had complicated discussions about a number of ab¬
struse points of logic, discussions which, though recorded, probably only ever attracted
the attention of a few scholars (for example. Logical Discussions; see [Mohaghegh and
Izutsu, 1974, pages 279-286]). By contrast. Logic for Samsaddln was read by nearly
every student. One may reflect that it was GazalT who had made its inclusion in the cur¬
riculum possible, even though the treatise deals with logical points that are almost entirely
absent from Gazall’s treatises. It is ironic that a scholar with interests confined to the ma¬
terial application of logic had done more than anyone else to find a permanent place for a
treatise dealing with entirely formal questions.
6 CONCLUDING REMARKS
These concluding remarks are, strictly speaking, more of an apology for the narrow focus
and relentlessly historical emphasis of this chapter, and for the tentative nature of its
claims. To trace a set of logical discussions from fragments written in Abbasid times
through to the introductory madrasa texts of the late thirteenth century entails finding a
topic common to all the texts. That common topic, the syllogistic, is apt for study due
to the existing scholarship. I further think (to repeat my introductory comments) that
tracing such a common topic is worthwhile above all because it begins the process of
delineating the framework of logical traditions which in turn determine the system that is
the primary object of discussion and dispute for any given logician. The very first thing
to do when setting out to study an Arabic logical work is to assign the work to its proper
systematic context, that is, the texts it addresses and the methods by which it engages with
those texts. I cannot say that Muslim scholars adhered to the traditions apparent in the
syllogistic when they contributed to other areas of logic, but I think it is likely that they
did, and that the historical account put forward in this chapter may serve provisionally for
the study of other debates.
Here then is a summary of the points made in this chapter regarding the syllogistic
traditions. The account differs most significantly from Rescher’s in speculating that Avi¬
cenna belonged to an existing tradition already fundamentally different from Alfarabi’s,
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and in rejecting his claims regarding a ‘Western’ school of logic [Rescher, 1964, chapters
five and six].
Syllogistic traditions Alfarabi was the first truly independent logician writing in Ara¬
bic. We may discern three factors in his writings. Above all, he was a product of late
Alexandrian Aristotelianism, and drew, however remotely, on the texts and techniques of
that tradition. These he modified in response to the fact that he was a Muslim scholar
working on a foreign and pagan intellectual tradition. Lastly, Alfarabi came to be con¬
scious of how badly the Aristotelian corpus was served by the interpretations in the exist¬
ing Alexandrian and Syriac works. This led Alfarabi to try to revive a true Aristotelian¬
ism after a period of rupture, which he did by writing commentaries on Aristotelian texts.
Though we no longer have the commentary on the first part of the Prior Analytics, we can
reconstruct his treatment of the conversion of contingent propositions from references in
the works of Avicenna and Averroes. In this treatment, Alfarabi tried to let the Aristotelian
text stand by finding an appropriate stratagem (in this case, a distinction prefiguring the
Averroist distinction between reading a term per se or per accidens). Alfarabi in his
moments of exegetical exertion was fairly dismissive of the ‘commentators’, presumably
members of the Syriac tradition from which he distanced himself.
Avicenna had, broadly speaking, the same philosophical ancestry as Alfarabi, and
claimed Alfarabi as his most eminent forebear after Aristotle. Avicenna’s prominence
among logicians in Iran and further east roughly parallels that of Alfarabi among the
Baghdad logicians. Many of his doctrines which seem idiosyncratic to us are not in the
writings of the Baghdad scholars, and are presented as though they are already known
to his readers. This procedure may indicate that he was simply modifying an existing
tradition, different from that of Baghdad. In any event. Avicenna’s syllogistic differed
radically from Alfarabi’s, and he set out some important points in his system by explicitly
stating what was wrong with Alfarabi’s corresponding doctrine. The single most impor¬
tant factor determining these differences was the fact that whereas Alfarabi thought that
the Aristotelian text would, with sufficient attention, yield a coherent system, Avicenna
thought that he already knew the coherent system, and used it to identify obscure parts of
the text. Alfarabi bent his system to the text, Avicenna bent the text to his system.
In his logical writings, Avicenna covered the same territory as Alfarabi’s Aristotelian
commentaries. Avicenna did not go on, however, to deal to the same extent that Alfarabi
had with the problems of relating logic to the Islamic disciplines. That strand in Alfarabi’s
logical writings was taken up by various Andalusian logicians, and by GazalT. GazalT
prepared the ground for the institutional acceptance of logic, a Farabian task, but he did
it by basing his formal treatment on the elementary section of Avicenna’s Philosophy
for ‘Ala’addawla. GazalT’s work on the syllogistic, however, was so superficial as to be
negligible.
GazalT is a special case, because he wrote primarily as a jurist and a theologian. But
by his death in the twelfth century, two logical traditions had emerged, one Farabian, the
other Avicennan. The finest representative of the Farabian tradition was the Andalusian
Averroes, who in his syllogistic developed doctrines found in Alfarabi’s writings. In fact,
Averroes tells us he took from Alfarabi the distinction between the per se and per acci-
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dens. But Averroes' relation to Alfarabi is complex. As Averroes developed the incipient
Aristotelianism of Alfarabi, he became increasingly less satisfied with the Farabian an¬
swers to exegetical problems, and sought more global solutions which gave every part
of the Aristotelian text due weight. This is an extension of the Farabian attitude to the
Aristotelian text, in which every position adopted is intrinsically defeasible in the face of
a better stratagem. The other scholar to whom Averroes made constant reference, aside
from Aristotle, is Avicenna: Avicenna had problematized the Aristotelian system, and
thereby determined those points on which Averroes had to dwell longest.
The other logical tradition, the Avicennan, had by the early twelfth century at the latest
come to identify problems and cruces in Avicenna's syllogistic which were to occupy
the tradition thereafter. Avicennan logicians ceased to consider anything other than the
system Avicenna had used in judging Aristotle’s logic and, though referring to Avicenna
generally as ‘the most eminent of the later scholars’, never treated his texts or doctrines
as immune to criticism and modification. Some representatives of the tradition, such as
RazT, were even fairly scathing about Avicenna’s expositions, though in the late thirteenth
century others began to refer reverentially to Avicenna and find ways to explain away his
logical errors. This did not, however, prevent them from modifying his logical system
exactly the same way as the earlier Avicennan scholars.
The twelfth century saw the clear delineation of the Farabian and Avicennan traditions,
each of which paid attention to the other’s founder, but rarely to his epigones. The twelfth
century also saw other writers referring to Alfarabi, Avicenna and Aristotle. These writers
in some cases tended more to an Avicennan systematic, such as Abu-l-Barakat al-Bagdadi,
in other cases, to a more Farabian, such as ‘Abdallatlf al-BagdadT. At least in the case of
the former, however, there is no evidence that his syllogistic was developed further by
his students, and it really speaks past the interests of the mainstream Avicennan tradition.
Neither scholar enjoyed a posterity. Even the Farabian tradition guttered, and after the
middle of the thirteenth century, the Avicennan tradition had come to predominate in the
Muslim world. The Farabian tradition had been weakened by its continued fixation on the
non-Muslim Aristotle; although Avicenna worked in conversation with Aristotle and the
later peripatetics, logicians after him worked directly on the system against which he had
measured Aristotle.
Avicenna was in one respect too successful in naturalizing the study of Aristotelian
logic. Though the Averroist approach was intrinsically less stable and mired in a Greek
past, it turned out to be transportable, because it spoke directly to the problematic of Latin
writers after the coming of the logica moderna. Avicennan logics by contrast were only
translated much later (see e.g. [Brockelmann, 1936-1949, Sup. vol. I, page 845]), and
aroused no interest. The problem was that it was no longer obvious which parts of the
Avicennan system were commensurable with the Aristotelian. It is a problem which still
plagues the study of his logic, and research needs to be directed to clearing the ground
preparatory to making such assessments.
The work ahead I hope that I have conveyed some sense of how many tasks await atten¬
tion in the study of medieval Arabic logic. Even in the narrow range of material examined
in this chapter, there is much to be done. There has been no sustained effort to reconstruct
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585
Alfarabi's modal syllogistic (though Lameer has announced he is preparing such a study);
there has been no plausible interpretation given of Avicenna’s modal syllogistic, and there
are still many problems in understanding his hypothetical syllogistic; we have no over¬
all picture of what Averroes was doing; and the other post-Avicennan logicians are, with
two exceptions among writers on the syllogistic, largely uncharted territory. And this is
in the syllogistic, one of the logical disciplines which has been relatively well treated in
the scholarly literature. This is not to say that there are not valuable studies in the other
logical disciplines; there are (see appendix three for lists of such work), but they suffer
from even worse gaps in coverage.
Of the many, many logical issues left out of consideration in this chapter, one is par¬
ticularly noticeable by its absence, and a few words are in order as to why. I have made
no attempt to identify the modal notions which lie behind the syllogistic systems of the
various authors. Averroes’ modal notions have been compared with the broader range
of options explored in the middle ages [Knuuttila, 1982, pages 352-353], and a lengthy
study has been made of Avicenna’s conception of the modalities [Back, 1992]. These are
valuable contributions. As a matter of procedure, however, I think the preliminary task
should be to lay out as precisely as possible the syntactic outline of any given system, and
only then investigate its underlying conceptions of modality.
The most unfortunate consequence of concentrating on the syllogistic, however, is that
it leads to minimizing consideration of how Islam influenced the constitution of logic in
its realms. It has been noted in the course of this chapter how apologetic tendencies drove
early logicians like Alfarabi to argue for logic’s utility for and complementarity with the
Islamic disciplines of grammar, theology and jurisprudence, and how later theologians
and jurists like GazalT came to accept these arguments. But there are many more, and
more complex, issues to take into account in studying the relation between Islam and
logic. The great historical task is working out the precise clashes between the Islamic
disciplines and philosophy which left the logical treatises as narrowly focussed as they
are, and finding the genres which took over treatment of these previously logical topics.
This is really the key point for future research, because as each book of the Organon
gave way to a competing Islamic discipline (as for example topics gave way to adab al-
baht, and rhetoric to 7/m al-ma'ani ), aspects of the original Aristotelian discipline either
transmuted or decomposed into other disciplines. And if ever we can appreciate those
changes, we can speak not only of the contributions of Muslim scholars to logic, but also
of the contributions of logic to Islamic culture.
ACKNOWLEDGEMENTS
I have incurred a number of debts in writing this chapter, and it is a pleasant duty to
acknowledge them here. Most of all, I am indebted to Dimitri Gutas, who kindly read
and commented on an earlier version of this chapter. His suggestions have saved me
from numerous mistakes and omissions. Henrik Lagerlund and Christopher Martin also
read an earlier version of this chapter, and I am grateful to them for their suggestions.
I regret that I was not able to deal comprehensively with a number of the points they
raised, and can only hope that future research will make studies on Arabic logic more
586
Tony Street
satisfying for those who work on Western logic. There are a number of other people who,
though not having read the chapter, have helped in its construction in one way or another.
They did so generally in the course of looking at earlier attempts I have made to study
Arabic logic, and making me aware of relevant manuscripts, studies I had overlooked,
or various mistaken conceptions. So, thanks to: Ahmad Hasnawi, Dominic Hyde, James
Montgomery, Ahmed al-Rahim, and David Reisman. Lastly, as with all who work in the
field of Arabic logic, I owe a vote of thanks to Nicholas Rescher, both for his pioneering
work, and for his generous words of encouragement.
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APPENDICES
A AVICENNA’S MODALS
Modalized propositions are represented (from left to right) by a modal operator, followed
by the predicate, the subject, and a superscripted letter indicating quantity and quality.
The modal operators are as follows: X stands for an absolute ( mutlaqa ) proposition, A
for a perpetual ( da’ima ), M for a possible ( mumkina ) and L for a necessary ( daruriyya ).
The default reading is date, wasfl readings are indicated by a superscripted w to the right
of the modal operator. Premise-sets are given in order of major, minor and conclusion (if
applicable). All references to Pointers may be checked in [Inati, 1981], which gives the
Arabic page numbers in the margin.
Purely datT premises
X\ A contradictories See [Avicenna, 197T 2 , pages 307-308].
X x bf £ Abj°
Xxbf £ Abf
Xx bf f Abj e
X\bj° £ Abj a
(Square for M\L is isomorphic; see [Avicenna, 1971 2 , pages 318-319].)
X 2 A contradictories See [Avicenna, 1971 2 , pages 309-311]
X 2 bj a £ Abf V Abj°
X 2 bj e £ Abf V Abj°
X 2 bf f Abf V Abf
X 2 bj° f Abj a V Abf
Arabic Logic
591
(Square for M 2 L is isomorphic; see [Avicenna, 1971 2 , pages 319-320].)
Perfect first figure mixes XXX, XLX, LXL, LLL, MMM, MXM, MLM. Proofs
for some second-figure moods also assume AX A and ALA. See [Avicenna, 1971 2 , pages
387-397],
X conversions X e-conversion fails. X\ a- and i-propositions convert as X\ i-propositions.
X 2 a- and i-propositions convert as X\ i-propositions. See [Avicenna, 1971 2 , pages 321—
333]
Substituting M\ for Xi and M 2 for X 2 gives all M conversions; see [Avicenna, 1971 2 ,
pages 338-340]
L conversions L e-proposition converts as L e-proposition. L a- and i-propositions
convert as M i-propositions. See [Avicenna, 1971 2 , pages 334-337]
Further development In the first figure, there are two imperfect mixes: LML, XMM.
See [Avicenna, 1971 2 , pages 391-395],
In the second figure, the following are proved: LLL, XLL, LXL, MLL, LML.
Premise pairs XX, MM, XM and MX all fail to produce. See [Avicenna, 1971 2 ,
pages 403-407]
In the third figure, the following are proved: XXX, LLL, LXL, XLX, MMM,
XMM, MXM, LML, MLM. See [Avicenna, 1971 2 , pages 423-426]
Wasfi premises
Avicenna introduces the wasfi as one of his stratagems (along with the temporal) to find
a proposition which will have a contradictory and a converse “in its own kind”. (The
temporal is only considered for contradiction and conversion, and for nothing else.) He
takes the wasfi e-proposition (say) to convert as a wasfi e-proposition, and to be contra¬
dicted by a wasfi i-proposition. He makes no mention of the temporality being different
between the contradictories, and refers to these wasfi propositions as ‘absolutes’. Pure
wasfi premises will produce all fourteen moods in the three figures.
In Pointers, Avicenna also investigates dati-wasfi mixes, proceeding from Barbara
L W LL.
If the major is absolute, and the time of its assertion is as long as the subject
remains described by whatever it is described by, then the conclusion will be
necessary, because J is B always, and it has been posited that B, as long as
it is B, is A; so J is always A—so here the conclusion is necessary and the
major absolute. [Avicenna, 1331 2 A.H., pages 57-58]
592
Tony Street
B LATER MODAL LOGIC
There is no purely dati logic among the later logicians, in that ddti premises in some cases
convert as wasfi propositions. XA contradictories are all as in Avicenna, and are still
isomorphic with the tables for ML propositions. Perfect first-figure mixes differ most
significantly in that all syllogisms with possible minors fail. X conversions are the same
as according to Avicenna, but M propositions all fail to convert. L conversions are all dif¬
ferent; an L e-proposition converts as A e-proposition, and L a- and i-propositions convert
as A w i-propositions. Temporals are treated extensively, but none matches Avicenna’s
use, nor do they work as assertorics.
The wasfi propositions are differentiated fully as to modality and temporality.
Absolute wasfi contradictories
A w bj a £ X w bj°
A w bj e £ X w bj l
A w bj i £ X w bj e
A w bj° £ X w bj a
Absolute wasfi conversions A w bj e converts to A w jb e \ A w bj a and A w bj l convert to
X w jb i .
Syllogisms The rule for productivity in the first figure is given as follows; note that it
includes the wasfi propositions:
(1) The minor premise must be one of the seventeen actuals.
(2) If the major is not one of (DC), (VC), (DC& ~ VC), then the mode of the
conclusion is that of the major.
(3) If the major is one of these four, then the mode of the conclusion is like
that of the minor except that
(a) the restriction of the conclusion is same as the restriction of the major
[and]
(b) the conclusion is necessitated if and only if both the minor and the major
are.
(4) All other moods are non-productive. [Rescher and vander Nat, 1974, page
36]
Rescher’s semantics
I give here the names of the propositions, with examples and symbolic rendition, due to
Rescher and vander Nat. It may prove a helpful reference for the propositions referred to
throughout the text. Note that we cannot be sure that all Avicennan logicians meant the
same thing by a given proposition. It is certain that Avicenna did not mean his proposi¬
tions to be taken this way.
Arabic Logic
593
Rescher and vander Nat begin their representation by putting forward R t as the ba¬
sic operator for realization-at-time-f (which is described more fully in N. Rescher and
A. Urquhart Temporal logic (New York and Vienna, 1971) at pages 31-32), and then use
it to make the following abbreviations:
TQ X = Rt(Qx)
SQ X = Rs{Qx)
3Q X = (3 t)R t {Q x )
VQ X = (W)Rt(Q x )
UTQ X = DR r {Q x )
aSQ x = oRs{Q x )
3 OQ x = (3t)aR t (Q x )
VD Q x = maR t (Q x )
OTQ x = OR r (Qx)
o SQ X = ORsiQx)
30 Q x = (3t)OR t (Q x )
\/OQ x = (Vt)OR t (Q x )
They go on to say:
In our symbolizations of modal propositions, we shall systematically sup¬
press the temporality condition (£) relation to the existence of the subject.
Concerning the symbolic rendition of modes, we take notice of the following
points. First, in adopting the symbolic machinery we have, we assume that
all the usual quantificational and modal rules hold. Secondly, in the £-modes
the existence condition has been suppressed; fully stated, (□£) (All A is B),
for example, would be (Vx)[(3f)/?t.A I D ( \/t)OR t (A x D B x )\. Thirdly, the
modes T and S are special time-instantiations, with regard to the existence
of the subject, and accordingly, we here use ‘T’ and L S' as time-constants.
[Rescher and vander Nat, 1974, page 32]
With these preliminaries in hand, they then go on to offer symbolic renditions of the
various a-propositions as presented in a late text [SirwanT, ] as follows ([Street, 2000]
gives all Arabic terms used to present and define these propositions, and in the order here
presented, though note that the translation there of wasfi as ‘composite’ is wrong, and I
would now adopt the Sprenger/Rescher terms for the propositions):
The propositions
1. L: Absolute necessary (□£):
(Vz)[3A, D VOB x ]
All men are rational of necessity (as long as they exist).
2. L w : General conditional (DC):
(Vx)[3A x DVO(A x D B x )]
All writers move their fingers of necessity as long as they write.
3. Absolute temporal (DT):
(Vx)[3A, D □ TB X \
The moon is eclipsed of necessity at the time when the earth is between it and the
sun.
4. Absolute spread (□£):
(Vx)[3A x D DSB X ]
All men breathe of necessity at some times.
594
Tony Street
5. A: Absolute perpetual (VC):
(V:r)[3A x D VB X ]
All men are rational perpetually (as long as they exist).
6. ■A"': General conventional (VC):
(Vz)[3A x D V(A X D B x )\
All writers move as long as they write.
7. X w : Absolute continuing (3C):
(Vt)[3A x d 3{A x D B x )]
All writers move while they are writing.
8. Temporal absolute (T):
(Vx)[3. A x d TB X \
All writers move at the time they are writing.
9. Spread absolute (S):
(Vx)[3A x DSB X ]
All men breathe at certain times.
10. X\: General absolute (3C):
(Vi) [3A, D 3 B x \
All men breathe (at some times).
11. Possible continuing (OC):
(Vx)[3A x D 3 0{A X D B x )]
All writers move with a possibility while they are writing.
12. Temporal possible {OT):
(Vi)[3A x D OTB x ]
The moon is eclipsed with a possibility at the time when the earth is between it and
the sun.
13. Mi: General possible (OC):
(Vz)[3A x D 3023 x ]
All writers move with a possibility (at some time).
14. Perpetual possible (OC):
(Vx)[3A x D OSB X ]
All men breathe with a possibility at all times.
15. Non-perpetual necessary (□£& ~ VC):
(Vi){3A x d [VO B x k ~ VB X ]}
16. Special conditional (DC& ~ VC):
(Vi){3A x D [VD(A X D B x )k ~ VB X ]}
17. Temporal (OT& ~ VC):
(Vx){3A x D [□TB I &~VJB I ]}
Arabic Logic
595
18. Spread (□£>& ~ V£):
(Vx){3A x D [□5B I &~VB X ]}
19. Non-perpetual perpetual (V£& ~ V£):
(Vz){3A x d [VB X &~VB X ]}
20. Special conventional (VC& ~ V£):
(Vx){3 A x D[\/(A x DB x )&c~VB x }}
21. Non-perpetual continuing absolute (3 Ck. ~ V£):
(V:r){3A x D [3 (A x kB x )k ~ VB X ]}
22. Non-perpetual temporal absolute (T& ~ V£):
(Va:){3A x D [TB X &~VB X ]}
23. Non-perpetual spread absolute (£>& ~ V£):
(Vx){3A x D [5B X &~VB X ]}
24. X 2 : Non-perpetual existential (3^& ~ V£):
(V*){3A a D [3B X &~VB X ]}
25. Non-necessary existential (3£& ~ Of):
(Va:){3j4 x D [3B x k ~ VDB X ]}
26. M 2 : Special possible (0£& ~ □£):
(Vx){3A x D [3OB x &~VDB x ]}
C BIBLIOGRAPHICAL NOTES
The best general introduction to the history of Arabic logic is still, sadly (given its age),
[Rescher, 1964]. All of the individual logicians listed in its concluding register demand
serious further study.
General bibliographical resources The best place to start for a comprehensive list
of logical studies is now [Daiber, 1999], updated against Index Islamicus and Bulletin
de philosophie medievale. The bibliographies of major medieval scholars are listed in
[Daiber, 1999], but note on Avicenna especially [Janssens, 1991]. A new bibliography
covering the articles, books and editions of more recent years is under preparation.
Terminology There is as yet no sure guide to the technical terms used by logicians
writing in Arabic; [Jabre et al., 1996] is extremely helpful, though has some limitations,
especially for terms relating to the modal syllogistic. [Endress and Gutas, 1992-] will ulti¬
mately provide the most important materials for a complete lexicon. Each sub-discipline
within logic has its own set of technical terms. The following works include valuable
glossaries: [Black, 1990; Shehaby, 1973b; Zimmermann, 1981]. [Street, 2000] is wrong
in translating wasfi as ‘composite’, but still gives important references that need to be
596
Tony Street
worked into any putative future lexicon for post-Avicennan usage. [Street, ] presents the
consecrated phrases by which logicians put forward propositions, proofs and so forth. In¬
dividual logicians occasionally have contingent or idiosyncratic usage. Thus especially
the early logicians tend to change terminology fairly readily [Lameer, 1994]. The israql
logicians (who worked in the tradition founded by the twelfth-century logician and meta¬
physician, SuhrawardT) had their own terms, a number of which are decoded in [Ziai,
1990].
Translation movement, and genres The translation of each work within the Organon
is treated in [Goulet, 1989-], though note the following important works which have come
out since its publication: [Black, 1991] for On Interpretation, [Hugonnard-Roche, 1999]
for demonstration, and [Aouad and Rashed, 1999] for the rhetoric.
The genres in which the logicians wrote have been studied in [Gutas, 1993], but this
study really stops at the fourteenth century, and many genres which should properly
should be thought logical have yet to be examined.
Short treatments (as for example on the heap and the liar paradox) have yet even to be
listed as they occur through the literature.
THE TRANSLATION OF ARABIC WORKS ON
LOGIC INTO LATIN IN THE MIDDLE AGES
AND THE RENAISSANCE
Charles Burnett
In the Middle Ages, and again in the Renaissance, several Arabic texts on logic
were translated into Latin. These included not only works by Arabic philosophers,
Avicenna, Algazel, Alfarabi and Averroes, but also texts originally written in
Greek, i.e. the Organon or corpus on logic by Aristotle, on which all Medieval
and Renaissance texts were ultimately based. 1 While one can understand how
Latin translations of Arabic works on mathematics, medicine, astrology and other
practical sciences could useful, it is more difficult to imagine how texts on logic
written in, and for, a Semitic language could make much sense in a language
which is completely unrelated to it. For Aristotelian logic is, of course, very
much language based. Moreover, while Latin scholars were lacking scientific texts
in mathematics and medicine, they already had good translations and detailed
expositions of at least the first half of the Organon (Aristotle’s corpus of logical
writings, with the Introduction— Isagoge —of Porphyry), made by Boethius in the
early sixth century. And, when they wished to complete the Organon, they were
able to do so by translating the texts directly from the Greek. In the mid-twelfth
century James of Venice is credited with the translating the Topics, the Prior and
Posterior Analytics with ‘authentic expositions’, and we have from the twelfth
century, translations from Greek of the Topics and the Posterior Analytics (twice).
And yet we find Gerard of Cremona translating the latter work from Arabic in the
same century, and in the thirteenth century all Averroes’ Middle Commentaries
on the Organon were translated. Why was there any need to do this? This is the
question that shall be addressed in this paper.
First, however, one should give some idea of the extent of the translation of logi¬
cal texts from Arabic into Latin (sometimes through the intermediary of Hebrew).
The earliest such translations were made in Toledo in the mid-twelfth century.
First, there are those of Gerard of Cremona, the doyen of the translators working
'For an overview of the Latin versions of Aristotle made in the Middle Ages, see B. G.
Dod, ‘Aristoteles Latinus’, in The Cambridge History of Later Medieval Philosophy, eds N.
Kretzmann, A. Kenny and J. Pinborg, Cambridge University Press, 1982, pp. 43-79.
Handbook of the History of Logic. Volume 1
Dov M. Gabbay and John Woods (Editors)
© 2004 Elsevier BV. All rights reserved.
598
Charles Burnett
in Toledo, who lived from 1114 to 1187. 2 He translated the Posterior Analytics ,
together with two texts related to the work: the commentary by Themistius, and
a work entitled ‘On the syllogism’ (the main topic of Aristotle’s work) by Alfarabi
(d. 950). Then, there are those associated with an archdeacon in Toledo Cathe¬
dral, Dominicus Gundissalinus, and a Jewish scholar called Abraham ibn Daud.
Ibn Daud came to Toleda as an exile from Islamic Spain in ca. 1160, and sought
the patronage of the Archbishop of Toledo that for translating the great philosoph¬
ical encyclopedia of Avicenna, the Shifa giving as a specimen translation of the
opening section ‘On Universals’. 3 His suggestion was presumably accepted, for we
soon find him collaborating with Dominicus on other parts of the Shifa including
the whole of the jumal on logic. Dominicus collaborated with another scholar —
John of Spain — in translating the ‘Aims of the Philosophers’ of Algazel. This
was largely derivative from another philosophical compendium of Avicenna, the
Danishpazeh. Avicenna’s and Algazel’s logical works were extremely popular in
the Islamic world, and had virtually replaced the original works of Aristotle on
which they were ultimately based.
In Cordoba in the late twelfth century, there was, however, an isolated attempt
amongst a small group of philosophers to return to Aristotle. Aberrant though this
was in the history of Islamic philosophy, its impact on Western philosophy was im¬
mense. For the group included Averroes (Ibn Rushd) whose Long Commentaries,
Middle Commentaries and Epitomes of Aristotle’s works started to become known
to Latin scholars within a few years of his death in 1198. Among these works were
the set of Middle Commentaries on the Organon (Talkhis al-Mantiq), which be¬
gan with a commentary on Porphyry’s Isagoge, and included (as was normal in the
Arabic context) the Rhetoric and Poetics . 4 The first three texts of these Middle
Commentaries — those on the Isagoge , the Categories and the De interpretatione
— were almost certainly translated by William of Luna in Naples, in the 1220s. 5
William of Luna is likely also to have translated the Middle Commentaries on the
Prior and Posterior Analytics. William must have had some connection with the
new university of Naples which had been founded by the Holy Roman Emperor,
Frederick II, in 1224. Frederick himself was very interested in Arabic logic, since
he persuaded the Mamluk Sultan to send one of the most distinguished Islamic
philosophers to his court in Palermo, Siraj ad-Din al-Urmawi, where ‘he wrote a
2 For the following paragraph I am indebted to H. Hugonnard-Roche, ‘Les oeuvres de logique
traduites par Gerard de Cremone’, in Gerardo da Cremona, ed. P. Pizzamiglio, Annali della
Biblioteca statale e libreria civica di Cremona, XLI, 1990, Cremona, 1992, pp. 45-56. See also
C. Burnett, ‘The Coherence of the Arabic-Latin Translation Program in Toledo in the Twelfth
Century’, Science in Context, 2001 (in press).
3 See D. N. Hasse, Avicenna’s De anima in the Latin West, London: The Warburg Institute,
2000, pp. 4-7.
^See D. Black, Logic and Aristotle’s Rhetoric and Poetics in Medieval Arabic Philosophy,
Leiden: Brill, 1990.
5 See Commentum Medium super libro Peri Hermeneias Aristotelis translatio Wilhelmo de
Luna Attributa, ed. R. Hissette, Leuven: Peeters, 1996, pp. l*-4*.
The Translation of Arabic Works on Logic...
599
book on logic for him’ (We do not know what this may have been). 6 After Fred¬
erick’s death in 1250, his son Manfred continued his intellectual interests, and, if
‘the translator of king Manfred’ is the same as Herman the German, one can see
a continuation of the project for translating Averroes. For Hermann the German,
on his return to Toledo, translated the commentaries on the Rhetoric and Poetics
in 1256.
The interest in Arabic texts on logic continued in Spain in the thirteenth century.
For the Dominican, who taught Arabic and Hebrew in the Dominican studium in
Barcelona, Ramon Marti, quotes (apparently directly from the original Arabic),
from Galen’s Book on Proof, and Averroes’s Commentary on the Topics, 7 and Ra¬
mon Llull, the indefatigable preacher and pamphleteer, wrote his own adaptation
(from the Arabic) of Algazel’s logic, in Catalan verse. 8
At the same time as William of Luna was translating Averroes into Latin,
Jacab Anatoli translated the Middle Commentaries on the Isagoge, Categories ,
De interpretatione, and Prior and Posterior Analytics into Hebrew. He too was
working in Naples, and specifically thanks Frederick II for his patronage. Anatoli
belonged to a family of Jewish translators, the Tibbonids, who translated other
texts of Averroes. This Hebrew tradition of Averroes’s works impinged on the
Latin tradition from the late fifteenth century onwards, when Hebrew texts of
Arabic works began to be translated into Latin. The culmination of this process is
represented in the most elaborate and ‘definitive’ edition of the works of Aristotle
in Latin, first printed with great pomp and ceremony by the Giunta brothers in
Venice in 1550-52, 9 and reprinted several times thereafter. Accompanying the
Latin texts of Aristotle were the commentaries of Averroes, as the title proclaims:
‘ Aristotelis omnia quae extant opera ... Averrois cordubensis in ea opera omnes qui
ad haec usque tempora pervenere, commentarii’ (‘All the extant works of Aristotle
... and all the commentaries on these works of Averroes of Cordova which have
survived to these times’). 10 To give an example of the richness of this publication,
one may list of the works included in the volumes on logic:
6 C. Burnett, ‘The “Sons of Averroes with the Emperor Frederick” and the Transmission of the
Philosophical Works of Ibn Rushd’, in Averroes and the Aristotelian Tradition, eds. G. Endress
and J. A. Aertsen, Leiden: Brill, 1999, 259-99 (p. 267) and D. N. Hasse, ‘Mosul and Frederick
II Hohenstaufen: Notes on Atlraddln Al-AbhariT and SiragaddTn al-UrmawT’, in Occident et
Proche-Orient: Contacts scientifiques au temps des Croisades, eds. I. Draelants, A. Tihon and
B. van den Abeele, Turnhout: Brepols, pp. 145-63.
7 The citation from the ‘Book on Proof’ occurs within a long passage translated by Marti
from ar-Razi’s Doubts on Galen, edited in C. Burnett, ‘Encounters with RazT the Philosopher:
Constantine the African, Petrus Alfonsi and Ramon Marti’, in Pensamiento medieval hispano:
homenaje a Horacio Santingo-Otro, Madrid: CSIC, 1998, pp. 974-92.
8 C. Lohr, ‘Raimundus Lullus’ Compendium Logicae Algazelis’, Ph. D., Freiburg im Breisgau,
1967 and id., l Logica Algazelis. Introduction and Critical Text’, Traditio, 21, 1965, pp. 223-90.
9 Aristotelis Stagiritae omnia quae extant opera... Averrois Cordubensis in ea opera
omnes. .. commentarii, ed. G. B. Bagolini, 11 vols, Venice: Giunta, 1550-52.
10 See C. Burnett, ‘The Second Revelation of Arabic Philosophy and Science: 1492-1562’ in
Islam and the Italian Renaissance, eds A. Contadini and C. Burnett, London: The Warburg
Institute, 1999, pp. 185-98.
600
Charles Burnett
1. Porphyry’s Isagoge, the Categories, and De interpretatione with the Middle
Commentary of Averroes and Levi Gersonides’s ‘supercommentary’, both
translated by Jacob Mantinus.
2. The Prior Analytics , with Averroes’s Middle Commentary as translated by
Johannes Franciscus Burana.
3. The Posterior Analytics, with Averroes’s Large Commentary in three trans¬
lations from Hebrew, those of Abraham de Balmes, Johannes Franciscus
Burana and Jacob Mantinus. These translations are set out in three paral¬
lel columns, up to the point where Mantinus’s ‘golden’ ( aureus ) translation
finishes rnorte preventus , and continues to the end in two columns, cover¬
ing in toto 1,136 pages! Also, Averroes’s Middle Commentary translated by
Johannes Franciscus Burana.
4. Averroes’s Epitomes and Questions concerning the whole of logic, translated
by Abraham de Balmes.
5. This is followed by an extraordinary series of letters on specific topics in logic,
attributed without reserve to Arabic authors, also translated by Burana:
Averroes, Epistola de primitate praedicatorum in demonstrationibus
Abualkasis Benadaris (i.e. Abu’l-Qasim ibn Idris), Quaesita de
notificatione generis et speciei.
Alhagiag bin Thalmus (i.e. Ibn Tumlus), Quaesitum
Abuhalkasim Mahmath ben Kasam (i.e. Abu’l-Qasim Muhummad
ibn Q& sim), Quaesitum
Abuhabad Adhadrahman ben Iohar (Abu ‘Abdarrahman ibn Jawhar
?), Epistole
6. The Topics and Sophistici Elenchi with Averroes’s Middle Commentaries
translated by Abraham de Balmes, and an incomplete translation of the
Middle Commentary on the Topics by Jacob Mantinus.
Thus we can see that there was considerable interest in Arabic logic, especially in
the court of Frederick II, and among Aristotelian philosophers in the mid-sixteenth
century. The Medieval translations of the Middle Commentaries of Averroes do
not survive in many manuscripts, with the exception of that on the Poetics, which
served instead of Aristotle’s original Poetics throughout the Middle Ages. Nev¬
ertheless, the manuscript evidence only partially reflects the popularity of a text.
For Roland Hissette, who has produced the most detailed edition of any of these
commentaries so far (that on the De interpretatione) has shown that, although
only three manuscripts survive, the work was used in 1229 by an early master
in the university of Paris, Iohannes Pagus, by two Danes also studying in Paris,
Martin and John of Dacia, and at least one anonymous writer; in the Renaissance
The Translation of Arabic Works on Logic...
601
its potential readership was large, since it was included in twelve editions of Aris¬
totle’s works printed between 1483 and 1560. 11 Moreover, references in Albert
the Great and brief surviving fragments show that, aside from Averroes’ commen¬
taries, Alfarabi’s summaries of at least the Categories , and the De interpretatione,
and his commentaries on the Prior and Posterior Analytics were known in Latin
in the Middle Ages. 12 Moreover, a summary of the Posterior Analytics had been
included in the Arabic encyclopedia known as the Brethren of Purity and was
translated into Latin with an attribution to Alkindi. 13 But the fact that these
logical texts had an Arabic origin caused problems to Latin scribes and readers.
First of all, it must be pointed out that, for the majority of translators in the
Middle Ages, including Gerard, Gundissalinus, and William of Luna, an extremely
literal translation of the original was the deliberate aim. The result was ‘barbaric
Latin’, as was frequently pointed out by Renaissance humanists. (Only Llull’s
poetic paraphrase of Algazel’s logic falls outside this extreme literality). Examples
of this ‘barbarous’ Latin are the use of ‘invenire’ (the root W-J-D; literally ‘to
find’) for ‘esse’ (‘to be’) — hence ‘inventum’ (literally ‘the found thing’) for ‘the
existent thing’ — and ‘intentio’ (‘ma‘na’, meaning both ‘meaning’ and ‘subject’)
for ‘thing’. 14 One may compare the translation of William of Luna with that of
Jacob Mantinus, where the relevant words are italicized: 15
William of Luna, ed. Hissette, p. 3: Et nomen et verbum similantur inten-
tionibus simplicibus, que non sunt vere neque false, et sunt ille que inveniuntur
preter divisionem et compositionem: verbi gratia: sermo noster ‘homo’ et ‘albedo’
quoniam, cum non coniungitur ei l invenitur' aut ‘non invenitur\ non est adhuc
neque verum neque falsum; sed significat quidem rem cui innuitur preter quod
disponatur res ilia per verum et falsum. Et propter hoc sermo noster ‘hyrococerus’
et ‘acnhagaribach’ non disponitur per verum neque falsum, dum non coniungitur
cum eo sermo noster ‘invenitur’ aut ‘non invenitur’, aut absolute aut in tempore,
11 Commentum Medium , ed. Hissette, pp. 4*-7* and 19*-24*.
12 M. Grignaschi, ‘Les traductions latines des ouvrages de la logique et l’abrege d’Alfarabi’,
Archives d’histoire doctrinale et litteraire du moyen age , 39, 1972, pp. 41-107.
13 Edited by A. Nagy in Die philosophischen Abhandlungen des Ja’qub ben Ishaq al-Kindi , in
Beitrage zur Geschichte der Philosophic des Mittelalters, 2, 1897, pp. 41-64.
14 A. Maieru, ‘Influenze arabe e discussioni sulla natura della logica presso i latini fra XIII
e XIV secolo’ in La diffusione delle scienze islamiche nel medio evo europeo , ed. B. Scarcia
Amoretti, Rome: Accademia nazionale dei Lincei, 1987, pp. 243-267.
15 Charles Butterworth translates the Arabic as follows: ‘The noun and the verb resemble
uncombined ideas which are neither true nor false, that is, the ones which are taken without
being combined or separated. An example of that is our saying “man” and “whiteness”. For
as long as “exists” or “does not exist” is not joined to it, it is neither true nor false. Instead
it signifies a designated thing, without that thing having truth or falsehood attributed to it.
Therefore, neither truth nor falsehood can be attributed to our saying “goat-stag” and “griffon”
unless “exists” or “does not exist” is joined to it — whether without qualification or according
to a particular time — and we then say “a goat-stag is existent”, “a goat-stag is not existent”
or “a goat-stag exists or does not exist’”: C. E. Butterworth, Averroes’ Middle Commentaries
on Aristotle’s Categories and De interpretatione, Princeton: Princeton University Press, 1983,
p. 126.
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Charles Burnett
et dicatur ‘hyrcocervus inventus ’, ‘hyrcocervus non inventus ’, aut ‘hyrcocervus
invenitur ’ aut ‘non invenitur ’.
Mantinus, ed. Giunta, I, fol. 68v: Nomen autem et verbum similia sunt rebus
simplicibus, quae neque verum neque falsum significant, eo quod sunt sine aliqua
compositione vel divisione, ut homo vel album, quoniam, si non additur ei 'est'
vel ‘non est', tunc nec verum neque falsum significat. Sed significat rem individ-
uam, sine tamen aliquo vero vel falso. Et ideo cum dicimus ‘hircocervum’ aut
‘chimeram’, neque verum neque falsum significamus, nisi addiderimus eis ‘est’ vel
‘non est' sive simpliciter vel secundum tempus, et dicamus ‘hircocervus est ’ vel
‘non est', vel ‘chimera fuit ’ vel ‘non fuit'.
Examples from Gerard of Cremona’s translation of the Posterior Analytics show
another characteristic of the Greek-Arabic-Latin transmission. Arabic cannot
form compound words. So hypothesis becomes ‘asl mawdu’ (‘placed root’), which
naturally becomes in Gerard’s translation ‘radix posita’, and enthymema becomes
‘qiyas mudmar’ ‘secret/covered syllogism’ which yields ‘syllogismus occultus’ (This
use of ‘syllogismus’ is obviously confusing). 16
As a result of the success of the translating-enterprise in Toledo, this literary
style, including the use of the same Latin translations of the same Arabic terms,
was employed for all translations from Arabic. It is evident that Scholastic philoso¬
phers of the Middle Ages were accustomed to the style and the peculiar meanings
of the words, to an extent that we find difficult to appreciate. Scholars such as
Albertus Magnus and Thomas Aquinas have a remarkably accurate understanding
of the doctrines of Averroes, Avicenna and the other Arabic philosophers, even
though they only knew them through Latin translations, and they would differ¬
entiate (for example) between the instances where ‘inventum’ meant ‘found’ and
where it simply meant ‘existing’, or ‘intentio’ meant ‘intentio’ or simply ‘a thing,
the subject’. Nevertheless, there are aspects of the Latin translation which would
have confused, or would have been unintelligible even to them. One is in the same
passage quoted above. For the mythical beast ‘hyrcocervus’ (‘goat-stag’) men¬
tioned in Aristotle’s text, Averroes, quite sensibly, added the nearest equivalent
in Arabic mythology: the “anqa’ mughrib’ — ‘the phoenix/griffon that excites
the curiosity’. William of Luna simply transiliterates this unintelligible word into
Latin (‘anchagaribach’ — in fact, suggesting that he read a variant not attested
in the Arabic MSS: “anqa’ gharlba’ — ‘the strange phoenix’), which soon became
corrupted in the Latin manuscripts and editions: ‘anchagaribach, anquaganba,
anquagauba, auquagariba etc.’ Mantinus, however, does for his Latin audience
what Averroes had done for his Arabic readers: he finds an equivalent which is
familiar to them, in this case, the chimera.
16 For more details see Hugonnard-Roche, ‘Les oeuvres de logique’, pp. 50-51.
The Translation of Arabic Works on Logic...
603
Another example where a literal translation from Arabic produced incompre¬
hension is a discussion of the use of cases in the noun: 17
William of Luna, ed. Hissette, p. 7: Et nomen etiam, cum genitivatur aut
accusativatur aut mutatur mutatione alia huiusmodi, non dicitur nomen absolute,
sed nomen declinatum... differentia est inter declinatum et non declinatum (et
illud est in casu ‘u’ in lingua arabica)...
Mantinus, ed. Giunta, I, fols. 69v-70r: Nomen preterea cum est in genitivo,
vel accusativo, vel alio casu, vel mutatur aliqua alia simili mutatione, tunc non
dicitur simpliciter nomen, sed nomen casuale... Interest tamen inter obliquum et
non obliquum nomen, ut in lingua Arabic patet ...
Here, William of Luna probably produced an accurate translation of the Arabic
text (‘the case (ending in) ‘u’), but this already confused the scribes, who wrote
‘in cau’ (vel. sim.), and Mantinus simply glossed over the phrase.
The problem with the Latin transmission of the Middle Commentaries of Aver-
roes in general can be summarised as follows:
Averroes’ intention in the Middle Commentaries is to paraphrase Aristotle’s
text (without directly quoting it), in a way that both brings out the logical se¬
quence of Aristotle’s arguments (hence his use of the ‘Porphyrian tree’ for the
arrangement of the subject matter in these commentaries), and makes the subject
matter intelligible to an Arabic audience. 18
The literal Latin translations of the Middle Commentaries make no concession
to their audience. In some Arabic-Latin translations the translator adds a marginal
gloss explaining the meaning of certain things specific to the Arabic language and
culture, while not changing the text itself. There is little evidence that this was
done by William of Luna.
The most obvious example of this mode of transmission can be seen in the case of
the Middle Commentary on the Poetics, which, as mentioned above, served instead
of Aristotle’s Poetics for the entire Middle Ages. Aristotle had included many
examples of Greek poetry to illustrate his text. Averroes systematically replaced
these examples with well-chosen illustrations from Arabic poetry. Hermann the
German, when translating Averroes’s text into Latin, did not substitute examples
from Latin poetry, but faithfully translated all the excerpts from Arabic poems
17 Butterworth (p. 128) translates: ‘Moreover, when a noun is put into the accusative or
genitive case or altered in some similar way, it is not said to be a noun in an absolute sense,
but an inflected noun. . .The difference between the inflected noun and the uninflected noun —
which, in the speech of the Arabs, is the noun in the nominative case (literally: the case ending
with ‘u’)... ’
18 See J. Puig Montada, ‘Averroes’ Commentaries on Aristotle: to Explain and to Interpret’, in
the proceedings of II commento filosofico nell’Occidente latino (saec. XIII-XV), Firenze-Pisa,
19-21 Oct. 2000 (in press).
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Charles Burnett
into Latin, thus unintentionally producing the only Latin anthology of Arabic
poetry in the Middle Ages. 19
Moreover, in respect to recovering the text of Greek works on logic, the Arabic
can be seriously misleading. For example, the Arabic texts of the Posterior Ana¬
lytics and the commentary on it by Themistius have been shown by Hugonnard-
Roche to be paraphrases of the Greek texts, and indeed it is this paraphrase that
was also used by Averroes. One result of this paraphrase was a distortion of Aris¬
totle’s own conception of the role of logic/dialectic in respect to the science. 20
According to the teaching of Aristotle, logic examines but does not prove the first
principles. The Arabic version of the Posterior Analytics that Gerard translated,
on the contrary, stated that the ars dialectica attempts to demonstrate the com¬
mon propositions in each science. So, given the ambiguities introduced by Latin
translations of Arabic logical texts in the Middle Ages, what were the reasons for
translating them in the first place?
The question can be answered most easily, perhaps, in the case of Gerard of
Cremona. In the context of Toledo in the twelfth and, indeed, thirteenth century,
there was no question of translating anything from Greek. Rather, Arabic culture
was so dominant, and so advanced in the area, that the task was simply to replicate,
as far as possible, that culture in Latin. Moreover, both Gerard of Cremona and
Gundissalinus had a model-curriculum on which to base their replication: i.e. the
Classification (or Enumeration) of the Sciences of Alfarabi, which both scholars
translated. Alfarabi not only provided a template for the subjects to be covered in
a course of ‘philosophy’ in the Aristotelian sense, but also referred to the textbooks
to be used in that course. The second chapter of Alfarabi’s book (after a chapter
on grammar) was on logic (‘dialectica’ Gerard; ‘logica’ Gundissalinus), and in it
he systematically went through the subject-matter of the Isagoge, the Categories ,
etc. finishing with the eight books of the Topics , and the Prior and Posterior
Analytics. For Alfarabi logic is a necessary propaedeutic to the other divisions of
philosophy dealt with in the work: Mathematics, Physics, Metaphysics and the
moral sciences. Gerard translated Arabic texts in all these subjects, and there is
good evidence that he taught ‘Arabic science in Latin’ (as one could say) in Toledo,
where he is referred to as ‘dictus Magister’-i.e. ‘the Teacher par excellence’.
Gerard would have known the necessity for logic as a basis for the study of
mathematics (in particular, geometry) from another text which he translated, and
which was well-known to his students ( socii ), who quote from him, namely Ahmad
ibn Yusuf ibn Ibrahim al-Daya’s Letter on Ratio and Proportion. 21 The subject
of the text is geometry, but ibn Yusuf starts with a long preface, taking the form
19 See W. F. Boggess, ‘Hermannus Alemannus’ Latin Anthology of Arabic Poetry’, Journal of
the American Oriental Society, 88, 1968, pp. 657-70.
20 Hugonnard-Roche, ‘Les oeuvres de logique’, pp. 52-4.
21 Ibn YDsuf lived in Cairo in the late ninth and early tenth century, and served the Tulunid
Sultans there.
The Translation of Arabic Works on Logic. ..
605
of a conversation between geometricians of different kinds, supposedly infront of
the Prince that he serves. The whole point of the conversation is to demonstrate
that logic is a necessary propaedeutic to geometry. Having practical knowledge of
mathematics, or knowing the theoretical texts off by heart, is not sufficient; one
must understand the principles of the art, which can only be gained by having
recourse to a higher art, namely logic. 22
It is in this context that one must see the endeavour of Gerard of Cremona
himself. He had no need to translate the texts of the ‘old logic’ which had been
known since the translations of Boethius, but he felt compelled to translate from
the available Arabic version the Posterior Analytics , especially since it was partic¬
ularly relevant to the arguments used in science, dealing with the different kinds
of syllogism, the rules for different kinds of argumentation, and in particular the
rules for demonstrative argument, whose importance in the work is indicated by
the Arabic title for the Posterior Analytics —‘ kita al-Burhan’ — for which Gerard
gave the literal translation De demonstrationibus. In addition, however, Gerard
translated, still from Arabic, the commentary of the late-fourth-century Greek
philosopher, Themistius, on the Posterior Analytics , 23 as a help for understanding
Aristotle’s notoriously difficult text. The bibliography of Gerard’s works composed
just after his death in 1187 also mention ‘Alfarabi De syllogismo’ , which has not
been identified in Latin, but is presumably Alfarabi’s commentary (or part of such
a commentary) on the Posterior Analytics. The priority of logic in a curriculum of
philosophy is further indicated by the fact that the socii of Gerard, in compiling
a list of his works after his death, put logic first.
For William of Luna, unfortunately, we can only guess why he undertook the
translation of Averroes’ Middle Commentaries, since he wrote no dedications, and
we have no references to his activities. All that I can suggest is that, after it
had become known to Jewish and Christian scholars that Averroes had para¬
phrased (in the Middle Commentaries) and written word-for-word expositions (in
the Large Commentaries) on the whole range of Aristotle’s works, these scholars
felt it important to put all the Commentator’s works into Hebrew and Latin. This
was appropriate, especially since the expositions of Aristotle of the ancient Greek
philophers were only known fragmentarily (Byzantine philosophers were, at the
same time as Averroes, attempting to fill the gaps), and Averroes was known to
use as the starting point of his own commentary, that of Alexander of Aphrodisias,
and also brought into discussion the comments of other Greek philosophers, such
as Themistius and Philoponus. Thus we can see the translation of Averroes’s
Middle Commentaries of the logical texts as part of a much larger enterprise, and,
indeed, the manuscript and printing edition of these works show that they trav¬
elled exclusively with other commentaries by Averroes. In addition to this, we can
22 See C. Burnett, ‘Dialectic and Mathematics according to Ahmad ibn Yusuf: A Model for
Gerard of Cremona’s Programme of Translation and Teaching?” in Langage, sciences, philoso¬
phic au xiie siecle , ed. J. Biard, Paris: Librairie philosophique J. Vrin, 1999, pp. 83-92.
23 A direct translation of this work from the Greek was not made until 1481.
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Charles Burnett
point to a particularly strong interest in Islamic logic in Frederick II’s entourage,
encouraged by the patronage and example of the emperor himself. 24
In the Renaissance, we can see an even greater desire for comprehensiveness,
when all the commentaries of Averroes on Aristotle’s Organon , and much Arabic
logic besides, were included in the Giunta edition of 1550-52. Only in recent times
has such an intensive interest in texts on Arabic logic been revived.
ACKNOWLEDGEMENTS
I am grateful for the advice of Dag Nikolaus Hasse and Dimitri Gutas.
24 See p. *** above.
INDEX
A (quantifier, all), 248
‘AbdallatTf al-BagdadT, 571, 579, 584
Abelard, 530, 551
AbharT (AtTraddln), 573, 576, 581, 582
Abu-l-Barakat al-BagdadT, 529, 539,
568-571, 573, 584
Abu-Bisr Matta, 531, 532, 534, 537,
555, 556
Abu-Nuh, 531
Abu-Sa‘Td as-Slrafi, 537, 555, 556
Abu-s-Salt, 561
Abu-Sulayman as-SigistanT, 532, 553,
554,561
Abu-Yahya al-MarwazI, 532
accent, 37, 84, 86, 95
accident, 37, 41, 84, 87, 96
Ackrill, J. L., 119
ad hominem, 48, 82
ad hominem proof, 81
affirmation, 120-122, 125, 132, 133,
149, 154, 156, 158
affirmation sentences, 159
affirmative sentences, 159
AI (antecedent interchange), 250
Alexander of Aphrodisias, 533-535,
542, 543, 548, 549, 554, 572,
578
Alexander the Great, 28
Alfarabi, 527-529, 532-545, 547-550,
553-555,559,561-566,568-
573, 579, 580, 582-585
Alums, 533
Allen, J., 36
ambiguity, 30
Ammonius, 533
amphiboly, 37, 84, 85, 95
analysis, 214, 241
Anaxagoras, 101
A (and), 249
answerer, 76
Ap-con (apodeictic conversions), 267
Ap-con(pa) (apodeictic conversions per
accidens ), 268
Ap-opp (apodeictic oppositions), 266
Ap-sub-a (apodeictic subalternations),
267
apodeixis, 108
Apology, 103
apophansis, 121, 126, 144
applied logic, 42, 45
argument, 115
argument conversion, 78
argument pattern, 115, 166
argumental conversion, 38
argumentation, 115
argumentational skill, 136, 137
Aristotle, 2, 9, 17, 19, 20, 24, 39, 40,
56,60,67, 72,81,91,93,97,
234, 235, 531, 533, 535, 536,
539, 542-545, 547, 549, 550,
552, 553, 555, 561-565, 567,
569-572, 575-579, 582-584
Aristotle’s dialectical treatises, 56
Aristotle’s fallacies, 57
Aristotle’s formal language, 128
Aristotle’s Lyceum, 32
Aristotle’s Thesis, 68, 69
arithmetic, 139
arithmetical demonstration, 139
artificial language, 143
AS (antecedent strengthening), 250
As‘arl, 557
h (assertion sign), 249
assertion axioms
A0-A4, 249, 265, 285
A5-A14, 265, 285
608
A15-A28, 285
A29, 299
assertion transformation rules
ArI(US), 266, 286
Ar2(MP), 249, 266, 286
Ar3(DDI), 249, 266, 286
Ar4(DN), 266, 286
ArI(US), 249
Athanasius of Balad, 530
Averroes, 526-529, 537, 542-545,549,
561-567,570,571,580,583-
585
Avicenna, 527, 528, 532-555, 558-
585, 590-592, 595
axiom, 139
axiomatic discourse, 136
axiomatic systems, 248, 279
axiomaticists, 105, 106
axiomatized deductive system, 105
BaqillanT, 557
Barbara, 116, 175, 186, 188, 210, 211,
214, 215, 234
Barbara (A3), 249
Barbara LXL (A5), 265
Barbara QQQ (A15), 285
Barbara XQM (A19), 285
Barhebraeus, 579
Barnes, J., 32, 39, 51, 55, 81, 220,
222
Baroco, 157, 177, 186-188, 210, 211,
215, 229
Baroco LLL (A9), 266
Baroco NNN, 53
base conditions, 271
for J-models for L-X-M (B1-B6),
272
for Q-models for QLXM' (BQ1-
BQ8), 294
for T3-models for L-X-M (Bl,
B2, BT3-B5), 278
basic principles of all being, 110
Becker, A., 260-262, 284
begging the question, 37, 84, 92, 96
belongs to every, 129
belongs to no, 129
belongs to some, 129
BIC (biconditional rule), 251
Bocardo, 157, 178, 186-188, 211, 215
Bocardo LLL (A10), 266
Bocardo NNN, 53
Bochenski, I. M., 104, 251
Boethius, 56, 81
Boger, G., 48, 97
Bolzano, 64
Bolzano’s Ableitbarkeit, 65
Bonitz, H., 75
Bryson, 140
Camestres, 177, 184, 186-188, 211,
215, 228
categorical sentence, 117, 129, 132,
142-144, 165, 166, 168, 172
Categories, 30, 31, 41, 115, 118, 121,
153, 159
category mistake, 152
CC (complementary conversions), 290
CC(pa) (complementary conversions
per accidens ), 290
cd (the contradictory of), 282
Celarent, 116,176,186,188, 210, 211,
214, 215
Celarent XQM (A20), 285
Cesare, 176, 186-188, 211, 215
Cesare LXL (A6), 266
chain condition, 279-282, 286, 305
challenge, 57
Cherniss, H., 34
Chrysippus, 543
Church, A., 106
classical, 59
Cohen, C., 85, 89, 93
combination, 155, 158
combination of words, 37, 84, 85, 95
common notion, 112
common notions in Elements, 113
common noun, 124
complete syllogism, 175, 218
609
completeness, 20, 113, 247, 278, 279,
281, 298, 299
compound statements, 94
computation, 38
Con (assertoric conversions), 250
Con (pa) (assertoric conversions per ac-
cidens ), 251
concludence, 191
concludent pattern, 116
concludent premiss pattern, 202
conclusion, 174
conditional proof, 47
connexive logic, 280
consequence of, 53
consequent, 37, 84, 89, 96
consistency, 113
contentious argument, 83
contentious argumentation, 140
contradiction, 230
contradiction error, 88
contradictories, 134-136,156,159, 231
contradictory, 51, 155, 162
contradictory sentence, 135
contraries, 135, 136, 147, 153, 154,
156
contrariety, 153, 230
contrary, 155, 162
contrary sentence, 135
contrasted instances, 237
conversion, 69, 220
conversion per accidens, 180
conversion per accidens rule, 181
conversion rule, 180
conversion rules, 166
Cooke, H. P., 119
Copi, I., 85, 89, 93
Corcoran, J., 31, 32, 34, 53, 55, 64,
73, 104, 105, 114, 149, 208,
234, 248, 279, 281
correlative, 155
counter-ex falso, 72-75
countermodels, 247, 257, 277, 289,
296, 299
Cresswell, M., 248, 251, 258, 260, 284
cut, 49
CW (consequent weakening), 251
-D-syllogism, 74
D-valid, 74
Darapti, 177, 186-188, 211
Darii, 116, 176, 186, 188, 201, 209-
211, 214, 215
Darii LXL (A7), 266
Darii QMQ (A29), 299
Darii QQQ (A16), 285
Darii QXQ (A18), 285
Datisi, 186-188, 211, 215
Datisi (A4), 249
DDI (definiens and definiendum in¬
terchange, 249
decision procedures, 247, 262, 263,
282, 283, 305, 306
declarative sentence, 120, 122
deducibility, 113
deducibility rule, 50
deducible from, 53
deduction, 107, 115, 226
deduction apparatus, 117
deduction rules, 178
deduction system, 166
deductionists, 105
deductive method, 108
definition, 15, 16, 41
demonstration, 5, 55, 108, 111, 139
demonstration error, 91
demonstration leading into an absur¬
dity, 228
demonstrative knowledge, 107
demonstrative science, 144
demonstrative syllogisms, 74
denial, 121, 122, 132, 149, 154, 156,
159
denotation, 155
derived rules for assertions
DRl(RV), 250
DR2(AI), 250
DR3(AS), 250
DR4(CW), 251
610
DR5(BIC), 251
DR6(SE), 252
derived rules for rejections
R-DRl(R-RV), 254
R-DR2(R-AI), 254
R-DR3(R-AW), 254
R-DR4(R-CS), 254
R-DRS(R-SE), 254
R-DR6(R-II), 255
determinate sentences, 159
dialectical argument, 35, 57
dialectical premiss, 57
dialectical problems, 35
dialectical proposition, 35
dialectical structures, 46
dialegestha, 57
direct, 52
direct deduction, 166, 229
direct proof, 227
direct syllogism, 53
Disamis, 178, 186-188, 211, 215
discovery, 2
distribution, 104
Ditisis, 178
division of words, 37, 84, 85, 95
DN (double negation), 266
does not belong to some, 129
Dorion, L.-A., 32, 83, 137
Dyonisius Thrax, 6
E (-i I), 249
Eaton, R. M., 104, 208
ecthetic proofs, 60
Eemeren, F. H. van, 57
Elements, 112
elenchus, 75
empeiria, 23
endoxa, 35, 36, 41, 93
entailment, 45
episteme, 23
epistemics, 107
equivocation, 30, 37, 84, 95
eristic argument, 75, 76
Euclid, 109, 112
Euclidean geometry, 17
Eudemus, 543, 572
Euler diagrams, 263
ex falso, 69, 71
ex falso quodlibet, 65
existential import, 150, 157
expert, 57
extension, 164
extra dictionem, 37, 87
extra dictionem fallacy, 84
extra-dictione, 88
fallacy, 33, 37, 46, 48, 77, 88, 96, 97,
115, 140
post hoc, ergo propter hoc, 89
ambiguity, 85
begging the question, 46, 91
many questions, 46, 93
noncause as Cause, 77
falsifying refutation, 82
falsity, 149, 151
Felapton, 177, 211, 215
Felpton, 186-188
Ferio, 116, 176, 186, 188, 209, 210,
215
Ferio LXL (A8), 266
Ferio XQM (A21), 285
Ferison, 178, 186-188, 211, 215
Festino, 177, 186-188, 210, 211, 215
first figure, 197, 209, 210, 214, 223
first-order logic, 59
follows from, 53
form, 18, 22
form of expression, 86
formal deducibility, 216, 226
formal language, 142, 143, 235
Formation rules
for L-X-M (FR1'-FR5'), 265
for Q-L-X-M (FR1'-FR5'), 285
for LA (FR1-FR3), 249
forms of expression, 37, 84, 86, 96
Forster, E. S., 32, 137
four-valued logic, 247, 259
Frede, M., 31
611
Frege, G. ; 22, 31, 67, 252
Frege-Russell logic, 105
Fregean, 56
Furley, D. J., 32
Gabbay, D., 97
Galen, 531-533, 543, 554
GazalT (Abu-Hamid), 555-559, 561,
579, 582, 583, 585
Geach, P. T., 247, 257, 261-265
generic syllogisms, 38
Gentzen conditions, 49, 71
Gentzen logic, 49, 50, 58, 59
Gentzen’s structural rules, 53
Gentzen, G., 49, 71
Gentzen-deducibility, 50
Gentzen-implication, 51
Gentzen-validity, 50, 59
genus, 41
geometry, 109, 112
Gersonides, 566
good argument, 47
good-looking argument, 47
Gorgias, 14, 29
Graham, D.W., 34
grammar, 23, 120
Green-Pederson, N. J., 34, 56
Grootendorst, R., 57
Hacker, E. A., 184
Hamblin, C. L., 33, 57, 86, 96
Hansen, H. V., 32, 33, 45, 77, 88, 90,
95, 97
Hempel, C., 2
Heraclitean Rule, 29
Heraclitus, 8, 29, 101
Hintikka, J., 32, 46, 57
Hippocrates, 101, 140
Hitchcock, D,, 48, 49, 64, 76, 97
Homer, 3, 86
Hughes, G. E., 248, 251, 258, 260
HunagT (Muhammad ibn-Namwar), 580,
581
Hunayn,529, 531-533
hypostatization of proof, 114
hypotheses, 52
hypothetical syllogism, 52, 71
^ (invalid), 272
(Q-invalid), 295
A-introduction rule, 40
i conversion rule, 182
I (quantifier, some), 248
Iamblichus, 533
Ibn-Bagga, 561, 562
Ibn-al-BihrTz, 531
Ibn-Haldun, 525, 538, 539, 556-559,
562, 574, 580, 581
Ibn-Hazm, 561
Ibn-Mallh ar-Raqqad, 567
Ibn-al-Muqaffa‘, 530, 533
Ibn-an-Nadlm, 533
Ibn-Nahma, 531
Ibn-as-Salah, 559
Ibn-Suwar, 532, 533, 535
Ibn-Taymiyya, 526, 560, 574, 581
Ibn-at-Tayyib, 531, 554
Ibn-Zur‘a, 532
-> (only if), 248
«■ (iff), 249
ignoratio elenchi, 37, 84, 88, 89, 96
Iliad, 3
immediate inferences, 266
imperfect syllogistic forms, 38
implication, 49
impossibility, 150
in dictione, 37, 84, 88
incompleteness, 20, 218
inconcludence, 188,191,193,194,196
inconcludent pattern, 116
inconsistency, 71
inconsistent sets of wffs, 305
indeterminate sentence, 147,159,194
indirect deduction, 229
indirect proof, 52
indrecti deduction, 166
induction, 107
inference, 17, 49
informal axiomatic system, 109
612
informal proof of invalidity
by contrasted instances, 255
from the ambiguity of a particu¬
lar proposition, 257
interrogative exchange, 57
intuitionistic deduction, 180
intuitionistic-like logic, 66
invention, 2
irreflexive, 50
Ishaq, 531-533
Isfara’InI, 557
Jaskowski, S., 53
Jaeger, W., 34
John Philoponus, 533, 535
Johnson, F., 252, 261, 265, 271, 277-
280, 282
Joseph, H. W. B., 104, 184
Kamaladdln ibn-Yunus, 573
Kant, I., 101
Kapp, E., 31
KasI (Afdaladdln), 568, 572, 573, 576,
577
KatibT (Nagmaddm), 529, 574-576,
581, 582
KattanT (Abu-‘Abdallah), 561
Keynes, J. N., 104, 208
Kilwardby, 566
Kindi, 529, 531, 535
Kneale, M., 3, 15, 31, 32, 264, 265
Kneale, W., 3, 15, 31, 32
knowing, 103
L (necessity operator), 265
A (necessarily), 262
law of non-contradiction, 145, 231
Lear, J., 31, 34, 52, 56
linear logic, 66
Locke, 81
Locke’s ad hominem, 76
logial consequence, 113
logic, 2, 3, 15, 18, 21, 23
logical, 31
logical consequence, 107, 230, 234
logical consequences, 216
logical constants, 129, 142-144, 146,
148
logical equivalence, 127
logical forms, 38
logical structures, 56
logical syntax, 53, 136
logically perfect language, 142
logically true conditional proposition,
105
logos, 29, 31, 121, 122, 144
Lukasiewicz, J., 31, 53,104, 208, 247-
249, 252,253,255-260,263-
266, 279, 280, 284-286, 295
M (-i L -i), 265
M (possibility operator), 285
Mahdl, 531
Maimonides, 566
major premiss, 174
major term, 174
Manders, K., 4
many questions, 37, 84, 96
material cause, 90
mathematical calculation, 107
mathematical necessities, 73
mathematical truth, 73
mathematics, 2, 21, 112
McCall, S., 68, 73, 247, 258-262, 265,
266,270-272,277-281,283-
290, 293, 303, 306
meaning, 145, 146
medical proof, 140
Meredith, C. A., 280
metalinguistic placeholders, 142
metalinguistic study of grammar, 118
metalogical deduction, 106, 187
Metaphysics, 102, 110, 113-115, 118,
142
Meth, 59, 69
method of completion, 179, 236
method of contrasted instances, 179,
188
method of counterargument, 191
613
method of division, 18, 20
middle term, 173, 174, 223, 225, 226
Mignucci, M., 34, 53
Mill, J. S., 86, 93
Miller, J. W., 104, 208
Min, 54, 56, 60-63, 70, 72, 75
minimality requirement, 65
minor premiss, 174
modal logic, 73
model, 117
models
J-models, 278, 279
Q-models, 294-299, 301, 302
T3-models, 278
modern symbolic logic, 21
modified method of contrasted instances,
192-194
modus ponens, 38, 53, 60, 69, 116
modus tollens, 38, 53
Modus Tollens (R-D), 253
monism, 23
monotonicity, 69-71
Moravcsik, J., 27, 97
MP (Modus Ponens), 249
Muhammad ibn-‘Abdun, 554
myth, 3
n+,272
n~,272
name ( onoma ), 41
natural deduction, 53
natural deduction system, 105
natural deduction systems, 248, 279
natural language, 119, 143
nature of definition, 16
necessary truth, 72, 74
necessitation, 58
necessitation because of, 70
necessitation from, 70
necessities, 71, 150
negation, 125, 158
negative sentences, 159
neopythagoreans, 31
neutrovalid, 116
Nicomachean Ethics, 102, 103, 107
no-retraction rule, 77
nomological necessity, 73
non-cause, 96
non-cause as cause, 37, 84, 89
non-classical, 59
non-existence, 159
non-falsifying refutation, 82
non-formal necessity, 73
non-logical constants, 143
Non-Circ, 54, 56, 60-64
nonlogical necessary truths, 71
nonmonotonic, 50
nonmonotonic logic, 66
nonmonotonicity, 65
Normore, C. G., 60
Nortmann, U., 260
1 (not), 248
noun, 119,120
O (-. A), 249
object langauge deduction, 106
object language, 143, 166
On Interpretation, 30, 31, 40, 115,
118, 120, 121, 124, 126, 127
On Sophistical Refutations, 27, 31, 33,
35,36,41,42,47,48, 76,81,
88, 96, 118, 136, 139, 142,
149
one-one principle, 39
one-premiss conversion rule, 166
one-premiss conversion rules, 180
one-premiss rule, 166
onoma, 41
ontology, 31
operational rules, 50, 59
Opp (assertoric oppositions), 251
opposite, 155
o (overlaps), 272
p (possibly), 262
paninvalid argument pattern, 202
paninvalidity, 196
pan valid pattern in Prior Analytics,
215
614
pan validity, 115, 180, 210
paraconsistent logic, 65, 66
paralogismos, 37, 85
Parmenidean Rule, 29
Parmenides, 9, 23, 29
Parry, W. T., 184, 208
Partative sentences, 159
partial attributive, 130, 169
partial privative, 130
partial sentence, 160
partial syllogisms, 209
pathological metaphysics, 29
pathological philosophy, 29
Patterson, R., 73, 260
Patzig, G., 31, 104, 208
PC (A0), 249
per impossible argument, 52
per impossible syllogism, 53
per impossible, 71
perfect difference, 153
perfect or complete syllogism, 112
perfect proof, 47
perfect rule, 47
perfectibility thesis, 47
petitio principii, 92
Phaedo, 103
phasis, 121
Philoctetes, 8
physicists, 31
Pinto, R. C., 88, 95
Plato, 5, 7, 9, 14, 18, 22, 24, 27, 41,
535, 555
Plato’s Academy, 32
Plato’s Euthydemus , 75
Platonic method, 18
plural propositions, 39
+ , 272
Porphyry, 533, 539, 572
Port Royal Logic, 104
possession, 155, 158
possibility, 150
possible, 150
possible worlds semantics, 260
Posterior Analytics, 31, 70, 110, 118,
124, 144
Powers, L., 29, 37, 97
predicable, 41
predicate term, 174
predication, 22, 225
Prem+, 55, 56, 60-63
premiss, 38
premiss-acceptability, 43
premiss-consequence, 43
premiss-pair pattern, 116
premiss-relevance, 43
premiss-selection rule, 77
principle of consistency, 230
principle of non-contradiction, 111
principles of deductive reasoning, 110
Prior Analytics, 31, 34, 38, 48, 54,
68, 71, 82, 117, 118, 121,
124, 148
Prior Analytics, 106-108, 179
Prior, A. N., 208, 283-285
privation, 155, 158
privative particular syllogism, 177
probative deduction, 166
problem, 41
problema, 133, 137, 214, 224
Prometheus, 101
proof, 5
proof ad hominem, 81
proofs by ecthesis , 261, 279, 280
proper restrictions, 60
proposition, 38, 39, 76, 88, 126
propositional logic, 105
propositions, 94
Protagoras, 29
protaseic argument, 40, 54
protasis, 38, 40, 123, 126
Pythagoras, 101
Pythagoreans, 31
q, 272
Q (contingency operator), 284, 285
Q-con (ordinary contingency conver¬
sions), 289
Q-inconsistent sets of wffs, 306
615
Q-sub-a (contingency subalternations),
289
Q-sub-o (contingency subordinations),
291
Q, 306
q+, 272
q-,272
quality, 104
quantity, 104
question-answer dialogue, 57
questioner, 76
questioning games, 33
Quine, W. V., 13, 67
QXQ-AAE-1 (A17), 285
R-AI (rejection by antecedent inter¬
change), 254
R-AW (rejection by antecedent weak¬
ening), 254
R-CS (rejection by consequent strength¬
ening), 254
R-D (rejection by detachment), 253
R-DDI (rejection by definiens and de-
finiendum interchange), 254
R-DN (rejection by double negation),
266
R-II (rejection by implication intro¬
duction), 255
R-RV (rejection by reversal), 254
R-S (rejection by Slupecki’s rule), 253
R-SE (rejection by substitution of equiv¬
alents), 254
R-US (rejection by uniform substitu¬
tion), 253
Ramsey, F., 22
RazT (Fahraddln), 529, 556, 557, 559,
568, 569, 571-581, 584
RazT (Muhammad ibn-Zakariya), 535
real definitions, 17
real-nominal distinction, 17
reckoning, 38
reductio, 228
reductio ad impossible , 80
reductio deduction, 166
reductio proof, 157
reductio proof, 210
reduction, 209, 211, 214
Ref, 77, 78, 80
reflexivity, 49, 69
refutation, 57, 75, 76, 81, 88, 90, 92
reinterpretation, 242
H (rejection sign), 249
rejection axioms
Rl, 253, 266
R2-R4, 266
rejection transformation rules
RrI(R-US), 253, 266
Rr2(R-D), 253, 266
Rr3(R-S), 253, 266
Rr4(R-DDI), 254, 266
Rr 5(R-DN), 266
relevance logic, 247, 280, 281
relevant logic, 65, 66
relevant premisses, 70
reputable opinion, 57
reputable premisses, 41
Rescher, N., 262
restriction, 54
restriction-conditions, 56
rhema, 41
Rhetoric , 149
Rist, J. M., 32
Robinson, R., 34
Ross, W. D, 255-257, 284, 289, 290,
303
Ross, W. D., 32, 81, 104, 208
Routley, R., 69
rule of deduction, 116
rules of inference, 12
RV (reversal), 250
54, 59
55, 59
S5 (system), 261
Salmon, W., 4
Sarahs!, 531
satisfaction-failure profiles, 63
616
Sawi (‘Umar ibn-Sahlan), 568, 572,
573, 575, 576, 581
schematic letters, 142, 143
scheme of argument, 30
scientific knowledge, 107, 114, 140
Scott, D., 34, 49
SE (substitution of equivalents), 252
Searle, J.R., 72
second figure, 201, 208, 210, 224
secundum quid , 37, 84, 87, 96
self-necessitation, 70
semantic necessities, 72
semantic rules, 275, 276
R^-AW, 276
R^-RV, 276
semantics, 53,113,145,162,163,166,
247, 255, 257, 260, 261, 271,
273
sentence, 120
sentence having a subject that does
not exist, 159
sentence having a subject that exists,
159
sequent calculus, 50
Sergius of Res‘ayna, 530
set-piece argument, 42, 43
Severus Sebokht, 530
Shoesmith, D.J., 66
simple conversion, 168
simple declarative sentence, 122
Simplicius, 533
single-premiss argument, 92
single-premissed syllogism, 55
singular sentence, 160
singular sentences, 159
SirwanT (Muhammad ibn-Faydallah),
575' 576
Smiley, T. J., 31, 53, 66, 104, 105,
114,208,248,252,258,279-
283, 305
Smith, J. M., 55, 56
Smith, R., 39, 40, 104, 105, 248, 257,
261, 279, 280, 284
Snell, B., 3, 5, 10, 11, 13, 14
Socrates, 28
Solmsen, F., 32
solo argument, 58
Sophist, 7, 9, 29
sophistical refutation, 36, 46, 57, 82,
83
sophistical refutations, 36
sophistry, 18
Sophocles, 8, 101
soundness, 113, 278, 289, 295, 299
speculative science, 31
standard treatment, 96
Steinthal, H., 23
Stephanus, 533
Stoics, 525, 526, 543, 544, 556
strategy, 30
Strawson, 23
structural rules, 59
Sub-o (subordinations), 268
subalterns, 220
subject matter, 30
subject term, 174
substitution, 242
Sugihara, T., 262
sullogismos, 108, 116
superfluous premiss, 61
superordinate argument, 49
superstructural conditions, 271
for valuations relative to J-models
(S1-S6), 272
for valuations relative to Q-models
(S1-S10), 295
for valuations relative to T3-models
(S1-S6), 278
syllogisity, 50, 51
syllogism, 38, 41, 46, 55, 90, 92, 104,
105, 116, 137, 225, 226, 235
syllogism in the broad sense, 38
syllogismos, 51, 53, 78
syllogisms
apodeictic, 247, 257, 282, 303
assertoric, 247, 257, 258, 260, 263,
265, 277, 280-283, 305
contingent, 247, 257, 283, 285,
617
286
syllogisms-as-such, 76
syllogisms-in-use, 76
syllogistic argument, 16
syllogistic deduction, 17
syllogistic deduction rules, 113
syllogistic deduction system, 107
syllogistic entailment, 60
syllogistic error, 88
syllogistic form, 106
syllogistic implication, 54
syllogistic logic, 10, 19, 21, 142
syllogistic rule, 47
syllogistic validity, 60
syllogisty, 65
syntax, 113, 231, 247, 255, 257, 263,
272, 273, 276
system S5, 261
systems
AP, 280, 282
G, 262-264
L-X-M, 247, 258, 260, 265, 269-
273, 277-279, 281, 283, 285,
286, 295
LA, 247, 257, 265, 266, 285, 295
LM, 247, 258, 259, 264
Q-L-X-M, 247, 258, 285-289, 303,
306
QLXM', 247, 288-290, 294, 295,
299, 302, 303, 305, 306
systems of strict implication, 59
Szabo, M. E., 49
tables
countermodels for LLL ... MLX
syllogisms, 273
countermodels for QMQ ... QQM
syllogisms, 299
countermodels for QQQ ... LQM
syllogisms, 289
McCall’s table 12 for Aristotle’s
judgments about the contin¬
gents, 289
tables for models
Mi-Mi, 273-276
M 6 -Mn, 296-298
Mis-Mie, 301-302
Tadarl, 533
Tahtanl (Qutbaddln ar-RazT), 526, 529,
568', 574
Tarski, A., 7, 64, 149, 234
techne, 23
term logic, 21, 142
the wise, 57
Themistius, 533-535, 572
Theodore, 531, 533
Theophilus of Edessa, 531
Theophrastus, 533, 543, 549, 550, 572,
578
theory of grammar, 40
theory of validity, 41
thesis, 51, 76, 88
thesis of propositional simplification,
40
third figure, 201, 208, 224
Thom, P., 31, 53, 261, 271, 277, 279,
284, 289
Thomason, S. K., 271, 276-278, 297
Thompson, M., 9
Thompson, P., 64
Timaeus, 5
Timothy, 531
topic, 30
topic-neutral, 36
Topics, 35, 41-43, 118, 136
traditional fallacies, 32
transitivity, 49, 69
truth, 149, 151
truth conditions, 144
truth preservation, 66
truth value, 143
truths of logic, 71
TusI (NasTraddTn), 529, 568, 569, 571—
579, 581, 582
two-premiss rule, 166
two-premiss syllogism, 188
two-premiss syllogism rule, 169, 183
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